Effectiveness evaluation method of constellation satellite communication system with acceptable consistency and consensus under probability hesitant intuitionistic fuzzy preference relationship

System effectiveness evaluation is an essential part of constellation satellite communication system research, with applications in project verification and optimization as well as tactical and technical measurement argumentation. This paper presents a systematic and comprehensive effectiveness evaluation method for a constellation satellite communication system under a probabilistic hesitant intuitionistic fuzzy preference relationship (PHIFPR), aiming to address the fuzziness and uncertainty in effectiveness evaluation. First, a proposed definition of PHIFPR describes the hesitancy of evaluators, provides hesitancy distribution information, and depicts the worst negative information and risk preferences in effectiveness evaluation. Then, we deduce the approximate consistency index of PHIFPR and establish a mathematical programming model to increase individual consistency when the approximate consistency index does not reach a predetermined level. In the sequel, a proposed group consensus index uses the PHIFPR-based Hausdorff distance to measure the closeness between evaluators’ judgments. Afterwards, a consistency and consensus improvement model is designed to retain the original opinions of evaluators to make the consistency and consensus of PHIFPRs acceptable. Moreover, a goal programming model is established to gain the approximate priority weights of schemes regarding the approximate consistency condition of a PHIFPR as a fuzzy constraint. Finally, an experimental example is offered to highlight the practicability and feasibility of the proposed method, and some comparative analyses with other methods offer insights into the designed method.


Introduction
Terrestrial cellular wireless networks have developed rapidly, making it easier for people to exchange information. However, due to geographical factors, about 98% of the earth's surface cannot be effectively covered. Therefore, no structure exists to transfer information within these regions. Fortunately, the constellation satellite communication system can effectively solve this problem, enabling seamless coverage of the earth's surface, anytime and anywhere, to complement the terrestrial wireless network. The constellation satellite communication system is characterized by complex technology, a long development period, and massive investment, involving many indicator parameter configurations, the constellation design scheme's strengths and weaknesses directly relate to mission application performance. If a designed scheme is wrongly chosen or cannot meet the application requirements, it will cause huge losses and affect military information system construction. Effectiveness evaluation can fully demonstrate how constellation satellite designed schemes meet application demand and determine the efficiency of schemes to provide an essential basis for their selection. At the same time, the pros and cons of designed schemes can be found, and decision-making information can be obtained to improve them and formulate schemes.
In summary, a valid and viable evaluation is a prerequisite to selecting a scheme and guiding its design, and a scientific method is needed to obtain an accurate evaluation. In the evaluation, multiple evaluators provide their judgments on a finite set of constellation satellite designed schemes and integrate their judgments into collective intelligence aiming to find a common solution; thus, it can be treated as a GDM, which is a powerful tool to address complex evaluation problems (Hochbaum and Levin 2006;Liu et al. 2019). In the past years, several scholars have conducted in-depth research on effectiveness evaluation methods based on the GDM theory to support the practical requirements of developing satellite communication systems. For example, Qin et al. (2015) proposed to evaluate space information applications based on the AHP, which can provide a reference for researching effectiveness evaluation. Zhang et al. (2016) aimed to evaluate military satellite communication systems using a dual hesitancy fuzzy value and index weights calculated by geometric means and least-squares. Liu et al. (2012) combined the unilateral chain and fuzzy evaluation methods to improve the validity and accuracy of an evaluation and provide a reference for effectiveness evaluation of satellite communication systems. To evaluate communication interference in a geostationary satellite orbit system layout, Dong et al. (2020) modeled interference estimation and analytical calculation under different scenarios and proposed an evaluation method based on the interference function extremum, which has certain reference value for communication interference evaluation and avoidance measure formulation. Based on the above literature review, most of the existing evaluation methods use a single precise value to model the uncertainty arising from human cognitive behavior. The satellite communication's related material and evaluators' knowledge are two primary information sources for evaluation in the actual evaluation process. The single precise value is often too restrictive and complex for evaluators to precisely express their individual preferences due to the inherent ambiguity and the complexity of the evaluation environment. Therefore, developing a powerful and effective method is necessary to provide a more logical, straightforward, and informative way of expressing information.
The evaluation process of the constellation satellite communication system usually requires evaluators to compare and rank the schemes. Preference relation (Altuzarra et al. 2010;Herrera et al. 2001;Negahban et al. 2017) is an effective tool to describe this evaluation problem, in which the evaluator can express their individual preferences by comparing each pair of constellation satellite designed schemes. Up to now, much attention has been paid to the study of its theory and application (Liao et al. 2020;Liu et al. 2020;Wang et al. 2020). Since traditional preference relations require evaluators to provide accurate values (Taninio 1984), however, the complexity of the environment and problems, as well as differences in thinking and knowledge, restrict the application of preference relations. To address this problem, scholars introduced Zadeh's (1965) fuzzy sets to preference relation to constructing fuzzy preference relations and generalized it in the evaluation situation. Furthermore, some extended forms of fuzzy preference relations have been successively proposed to allow evaluators to comprehensively and intuitively describe their individual preferences information from multiple dimensions, including interval-valued fuzzy preference relationship (Bustince and Burillo 2000), triangular fuzzy preference relationship (Xu 2002), incomplete interval fuzzy preference relationship (Xu 2007a;Xu et al. 2014), hesitant fuzzy preference relationship , and incomplete hesitant fuzzy preference relationship (Khalid and Beg 2017).
In the above-mentioned fuzzy preference relationships, the basic element of fuzzy preference relationships only provides the membership degree, which represents that a designed scheme is preferred over another, but ignores the corresponding non-membership degree. In addition, the hesitant fuzzy preference relationship can describe the membership degree of an element by a finite set of all possible values, but it overlooks considering the occurrence possibilities of every uncertain element. To deal with these issues, the IFPR was introduced by Xu (2007b), which makes the results of the preference process exhibit the characteristic of affirmation, negation, and hesitation by simultaneously considering the membership degree and non-membership degrees. In the real evaluation, evaluators cannot give explicit preferences over the scheme due to the complexity of the evaluation problem, it is suitable to make evaluators express their evaluation information from three aspects: "preferred," "not preferred" and "indeterminate," aiming to improve the accuracy of expressing a preference for the scheme. At present, many scholars devoted IFPR to group evaluation (Gong et al. 2018;Ma and Xu 2018;Xu and Liao 2015;Yang et al. 2019). And the PHFE was first proposed by Zhu (2014), as an extension of the hesitant fuzzy element (Torra 2010), which describes not only the hesitancy of evaluators when making the evaluation, but also the hesitancy distribution information. Because the PHFE is an important extension of the HFE, it is natural to develop the PHFPR and apply it to the evaluation process (Li and Wang 2018b;Lin et al. 2020;Liu et al. 2021). To more fully simulate human perception and cognition, this study synthesizes the advantages of the above two preference relationships by cultivating a more general fuzzy preference relationship, called a PHIFPR, which is more convenient and flexible than current fuzzy preference relationships and can more accurately reflect the evaluators' individual preferences by considering three aspects of the judgment and the occurrence probabilities of the uncertain elements.
In addition, the consistency index of preference relationship plays a crucial part in effectiveness evaluation, as it is concerned with the reasonableness of the result, which can be lead to unreliable decision results if the evaluation information does not meet the consistency requirement (Jin et al. 2020b). However, it is difficult to obtain a fully consistent preference relationship, in this circumstances, evaluators can strive for seeking an admissible consistent preference relationship to guarantee the rationality of evaluation result. With the growing complexity of the effectiveness evaluation problem, requiring an evaluator to consider all aspects of an evaluation is becoming impossible; therefore, there is always more than one evaluator making an evaluation. Since each evaluator has a different social experience and cultural background, individual opinions of evaluators can vary widely, leading to conflicts among them. Hence, group consensus is another critical issue in effectiveness evaluation, which is crucial in generating satisfactory solutions . To solve the effectiveness evaluation problem of constellation satellite communication systems based on individual consistency and group consensus involves three processes Zhang and Wang 2019): (1) checking the individual consistency, i.e., when the consistency level has not yet been achieved, individual consistency must be improved; (2) judging the group consensus among evaluators, i.e., if it cannot satisfy the requirement, group consensus must be improved so as to select the scheme with the highest satisfaction as much as possible; and (3) determining the priority weight of scheme based on evaluation information that meets individual consistency, and the ranking order of schemes can be generated by the priority weight of the scheme. The improvements of individual consistency and group consensus are most important among the above three processes, which is a challenging task for many researchers. To date, a large number of scholars have conducted fruitful research on the topic of evaluation problems based on individual consistency and group consensus (Jin et al. , 2020aLi and Wang 2018a;Tian et al. 2021Tian et al. , 2019Wu and Tu 2021;Wu and Xu 2016;Zhang and Pedrycz 2019). Generally, the individual consistency and group consensus improvement processes usually involve either mathematical programming modeling or iterative algorithms, and the former can save time and can be more practical in an emergency evaluation. From this point of view, we build a mathematical programming model to improve individual consistency and group consensus and shorten evaluation time in this paper.
The effectiveness evaluation of constellation satellite communication systems is influenced by many criteria, including coverage, quality of service, adaptability, interrupt performance, and protection performance. The description of some criteria may be uncertain and difficult to quantify, which directly intensifies the uncertainty and hesitation of the evaluation team in the evaluation process. Therefore, it is necessary to determine the optimal constellation satellite designed scheme by selecting an appropriate preference relationship description and establishing a suitable evaluation method. This paper focuses on an effectiveness evaluation method with acceptable consistency and consensus under PHIFPR, which can provide technical support for the development of the constellation satellite communication system. The main novelties and contributions of the proposed method are highlighted as follows.
1. We define PHIFPR, which expresses evaluators' views, includes the hesitancy of evaluators and hesitancy distribution information, and depicts the worst negative evaluation information and the evaluator's risk preference, which can be used to describe the evaluation information as completely as possible to avoid substantial information loss. 2. A Hausdorff distance formula, which does not require normalization, is designed to calculate the distance between two PHIFNs. The Hausdorff distance can effectively determine the distance between any two PHIFNs with different numbers of elements. It is not influenced by the subjective preferences of evaluators. 3. We define an approximate consistency matrix and appropriate consistency index to measure the evaluation information's rationality given by evaluators and define a group consensus index to measure the closeness between evaluators' individual opinions. 4. Several programming models are formulated to revise PHIFPRs with unacceptable approximate consistency and group consensus and determine the approximate priority weights of schemes according to the revised PHIFPRs. These models have a uniform mathematical structure to limit the amount of calculation.
The rest of this article is organized as follows. Section 2 reviews basic concepts of the IFPR, HIFS, and PHFPR. The PHIFPR and Hausdorff distance between any two proba-bility hesitant intuitionistic fuzzy elements are defined in Sect. 3. In Sect. 4, the approximate consistency matrix, approximate consistency index, and group consensus index of PHIFPRs are defined, and some programming models are designed to improve the individual consistency and group consensus among evaluators. In Sect. 5, we develop a modeling method to derive priority weights of schemes based on repaired PHIFPRs. Section 6 introduces a method to address the effectiveness evaluation problem of constellation satellite communication system with acceptable consistency and consensus under PHIFPRs. Section 7 describes an experiment to verify the practicality of the proposed method by ranking the constellation satellite designed schemes, and some comparisons are made to current evaluation methods. Some conclusions are presented in Sect. 8.

Preliminaries
This section recalls several basic concepts, such as the IFPR, HIFS, and PHFPR.
Definition 1 (Szmidt and Kacprzky 2003) Let Y = {y 1 , y 2 , . . . , y n } be a fixed nonempty universe set, the IFPR is denoted by a matrixÃ = (ã i j ) n×n , whereã i j = (y i , y j ), μ(y i , y j ), ϑ(y i , y j ) (i, j = 1, 2, . . . , n). For convenience, letã i j = μ i j , ϑ i j , where μ i j is the degree to which object y i is preferred to object y j , and ϑ i j is the degree to which y i is not preferred to y j , with conditions The elements of IFPR are ILVs, which consider both the membership degree μ i j and non-membership degree ϑ i j . Evaluators are often too restrictive and provide precise (crisp) evaluation opinions with difficulty for some schemes due to incompleteness or lack of evaluation information. Zhou et al. (2015) proposed the hesitant intuitionistic fuzzy set to overcome this drawback.
Definition 2 (Zhou et al. 2015) Let Y = {y 1 , y 2 , . . . , y n } be a fixed set, the HIFS h on Y is where μ h i (y i ) and ϑ h i (y i ) are, respectively, the membership and non-membership degrees of the element y i to 1] (l = 1, 2, . . . , L h i ), and ϑ h i (y i ) = 1 − max{μ h i (y i )} represents the maximum risk and the worst negative information on the basis of the greatest membership degree.
Definition 7 (Li and Wang 2018a) Let Y = {y 1 , y 2 , . . . , y n } be a fixed set. A PHFPR is denoted by a matrixH = (h i j ( p i j )) n×n , whereh i j ( p i j ) = {r l i j ( p l i j )|l = 1, 2, . . . , Lh i j } is a PHFE, indicating all possible degrees of y i preferred to y j , with conditions: where r l i j is the lth possible value inh i j , and p l i j is the corresponding probability of r l i j .

Probabilistic hesitant intuitionistic fuzzy preference relationship
The IFPR can describe the negative information in effectiveness evaluation, and the PHFPR considers hesitancy distribution information given by evaluators. The PHIFPR synthesizes these two preference relationships to more precisely describe evaluators' preference information. In this paper, we introduce a simplified PHIFPR based on the definition of the HIFS (Zhou et al. 2015) to avoid providing non-membership information in the complex evaluation process. In the simplified form, the non-membership degree is obtained from the maximum membership degree, which denotes that the non-membership degree will be an extreme value.
Definition 8 If Y = {y 1 , y 2 , . . . , y n } is a fixed set, then the PHIFPR is defined by a matrixQ = (q i j ) n×n , whereq i j = (y i , y j ), r (y i , y j ) p(y i , y j ), 1 − max{r (y i , y j )} (i, j = 1, 2, . . . , n). For convenience, letq i j = r i j ( p i j ), ϑ i j , where r i j ( p i j ) = {r l i j ( p l i j )|l = 1, 2, . . . , L r i j } denotes the possible priority degrees to which the object y i is preferred to the object y j , and ϑ i j = 1 − max{r i j } is the degree to which the object y i is not preferred to the object y j , with conditions: From Definition 8, we find that the PHIFPR, which considers the worst negative information and represents the risk preference by the non-membership degree, is composed of PHIFNs.
Then the operation laws of the PHIFNs are defined as follows, .
Its score function can be represented as Its accuracy function can be represented as We define a new comparison method for PHIFNs based on Definitions 10 and 11.

Definition 12
. . , L r 2 } be two PHIFNs. Assume all probability hesitant membership degrees in PHIFNs are arranged in ascending order, and let r be the jth values in q 1 and q 2 , respectively. We define the following comparison method.
, where at least one of "≤" should be "<," i.e., the equal signs of these inequalities cannot exist at the same time ( j = 1, 2, . . . , l; l = min{L r 1 , L r 2 }); 2. q 1 ≺ q 2 if q 1 < q 2 is not satisfied and S(q 1 ) < S(q 2 ), or if q 1 < q 2 is not satisfied, S(q 1 ) = S(q 2 ), and A(q 1 ) < A(q 2 ) (where "≺" means "inferior to"); The Hausdorff distance (Hung and Yang 2004;Huttenlocher et al. 1993), which is the maximum distance from a set to the nearest point in another set, is a reliable way to calculate the distance between sets. We define the Hausdorff distance of PHIFNs as follows.
The Hausdorff distance between q 1 and q 2 is Theorem 1 Let Q be a set of PHIFNs. The Hausdorff distance between PHIFNs has the following properties: Proof The proof of Theorem 1 is proved in "Appendix."

Consistency and consensus-improving models for PHIFPR
As is well known, the consistency of preference relation reflects the strict logical relations among each evaluation entry. The idea of fuzzy set theory is that everything is elastic to some extent (Zhao et al. 2021). Thus, the logic of strict consistency is incompatible with the uncertainty of fuzzy sets, which means that the concept of consistency is not applicable for PHIFPR. In this section, the approximate consistency matrix and approximate consistency index are first defined, and the corresponding mathematical programming model is built to improve the consistency level when PHIFPRs are not acceptably approximate consistency. Then, the consensus index of evaluators is used to check whether the group consensus level is achieved. A mathematical programming model is built to repair PHIFPRs when they are at an unacceptable consistency or consensus level.

Consistency-improvement process of PHIFPRs
This subsection first defines the approximate consistency matrix, and then the concept of approximate consistency index is introduced to test the consistency level of a PHIFPR based on the approximate consistency matrix and the additive transitivity. In addition, we devise a programming model to repair PHIFPR that is not acceptably consistent.
, and a i j denotes the smallest integer greater than a i j .
(2) ifq i j ≺q ji , which represents that scheme j is superior to scheme i, then eq i j < 0.5 and eq ji > 0.5. (3) ifq i j =q ji , which indicates no difference between schemes i and j, then eq ji = eq i j = 0.5.
From the analysis above, we can find that the approximate consistency measure matrix EQ = (eq i j ) n×n is a FPR. Consistency analysis in preference relation is the key to avoid evaluators not providing their opinions randomly and obtaining reliable results. Taninio (1984) defined the additive transitive property of a FPR as follows, (2) If the above additive transitivity is satisfied, we can infer that EQ is a consistency FPR.
Definition 15 SupposeQ = (q i j ) n×n is a PHIFPR whose approximate consistency measure matrix is EQ = (eq i j ) n×n .
EQ is an additively consistent FPR if and only if: (3) Given that the approximate consistency measure matrix EQ = (eq i j ) n×n is a FPR, we have eq i j = 1 − eq ji , i.e., the lower triangle elements of EQ can be deduced from the upper triangle elements of EQ, and we can write Eq. (3) as Due to the limited rational behavior of evaluators together with the complexity of the evaluation scenarios, the inconsistency in the evaluation information given by evaluators is usually existing. In the following, the approximate consistency index for PHIFPR is defined to measure the consistency level of the PHIFPR.
It is well known that the lack of consistency easily leads to inconsistent conclusions. Therefore, it is necessary to construct a mathematical programming model to improve the consistency level in the effectiveness evaluation (Jin et al. 2020a).
. . , L r i j }, and its approximate consistency measure matrix is EQ = (eq i j ) n×n . IfQ = (q i j ) n×n is unacceptable, then a mathematical programming model is constructed to . . , L r * i j }, and its approximate consistency measure matrix E * Q = (eq * i j ) n×n can be calculated according to Definition 14. The programming model is constructed to minimize the necessary adjustment, as follows: Let CI 0 be a preset threshold, which can be determined according to the situation and evaluators' preferences. How to determine an appropriate threshold requires further research. Saaty (1980) and Cutello and Montero (1994) conducted indepth research on this topic.
In Model 1, the first constraint is the approximate consistency index from Eq. (5), which ensures thatQ * satisfies the acceptable consistency condition. The remaining constraints ensure thatQ * is a PHIFPR. An acceptably consistent PHIFPRQ * can be obtained from Model 1 and applied to improve the consistency level of a PHIFPR.
We simplify Model 1 by removing the absolute value in the objective function and first constraint. We adopt the following notations. Let . Then we use |ζ i j | = ζ + i j + ζ − i j to remove the absolute value condition from the objective function. Letting , we can use |δ i jκ | = δ + i jκ + δ − i jκ to remove the absolute value from the first constraint to obtain Model 2,

Consistency and consensus-improving process of PHIFPRs
Evaluators can have diverse experiences and knowledge that causes their evaluation information to differ. The purpose of effectiveness evaluation is to balance their decision preferences and find a viable and acceptable scheme for the whole group. We introduce a group consensus index that can reflect the closeness degree among evaluators, using the Hausdorff distance to check the group consensus level. We also design a mathematical programming model to increase group consensus and individual approximate consistency when these are unacceptable.

Consensus index of PHIFPRs
Suppose the consensus index among k evaluators is calculated according to Eq. (6). The similarity degree SIM(e m1 , e n1 ) between any two evaluators e m1 and e n1 is obtained by the Hausdorff distance in Definition 13. The proximity degree PRO(e m1 ) between evaluator e m1 and other evaluators can be obtained based on the similarity degree SIM(e m1 , e n1 ), and the group consensus index CON(e) among {e 1 , e 2 , . . . , e k } based on the proximity degree can be obtained.
are the evaluation opinions of two evaluators. To capture the agreement degree between each pair of evaluators, we first define the similarity degree between them. The smaller the distance between e m1 and e n1 , the more similar they are. From this, we can define the similarity degree as, Let k n1=1,n1 =m1 SIM(e m1 , e n1 ) be the overall similarity degree associated with e m1 and other evaluators. Then PRO(e m1 ) (m1 = 1, 2, . . . , k) can be calculated between e m1 and other evaluators as, The sum of proximity degrees of all evaluators to all other evaluators is k m1=1 PRO(e m1 ). The consensus index CON(e) among all evaluators {e 1 , e 2 , . . . , e k } can be defined as, Since SIM(e m1 , e n1 ) = SIM(e n1 , e m1 ), we can simplify the above formula to It is obvious that 0 ≤ CON(e) ≤ 1, and the larger the consensus index CON(e), the greater the consensus among PHIFPRs. Moreover, CON(e) = 1 if and only if e 1 = e 2 = . . . = e k , i.e., when the PHIFPRs given by evaluators are all the same, which indicates full consensus. If CON 0 is a predefined consensus threshold, then the evaluation information of all evaluators has an acceptable consensus level if CON(e) > CON 0 . Otherwise, it must be improved.
Next, we next build a mathematical programming model to increase group consensus and approximate consistency when these are unacceptable.

New goal program to improve approximate consistency and group consensus of PHIFPRs
We design a mathematical programming model to improve individual approximate consistency and group consensus in effectiveness evaluation. SupposeQ m1 = (q m1 i j ) n×n (m1 = 1, 2, . . . , k) represent some PHIFPRs obtained from k evaluators, wherẽ q m1 i j }, must be obtained by the mathematical programming model. The approximate consistency measure matrices ofQ m1 = (q m1 ) n×n are E m1 The first constraint guarantees that the obtained preference relationship is of acceptable approximate consistency, the second constraint guarantees that the obtained preference relationship can reach the group consensus level, and the remaining constraints ensure thatQ m1 (m1 = 1, 2, . . . , k) are PHIFPRs.
We remove the absolute value condition from Model 3 to simplify the presentation. Let η m1 i jκ = 0, (i = 1, 2, . . . , n − 2; i < j < κ; m1 = 1, 2, . . . , k), Due to the complexity of Model 4, directly using MAT-LAB can reduce the accuracy of results. The GA is a class of intelligent optimization algorithms proposed by Holland Pal and Bhandari (1994), which is a stochastic search and optimization technique guided by the principles of evolution and natural genetics with substantial implicit parallelism. Drawing on the theory of biological evolution, the GA models the problem as a biological evolutionary process. It generates the next generation of solutions through selection, crossover, mutation, etc. And it gradually eliminates the solutions with low fitness function values and adds the solutions with high fitness function values. In this way, it is possible to evolve individuals with high fitness function values after multiple generations. Thus, this paper adopts the GA to solve Model 4 to obtain the modified PHIFPRs with acceptable consistency and consensus.

Deriving approximate priority weights from PHIFPRs
To obtain the reliable ranking of the schemes eventually, how to derive the priority weights is also a hot issue in effectiveness evaluation (Jin et al. 2020b). If the approximate consistency of PHIFPR given by the evaluator is unacceptable, then we cannot guarantee the rationality and reliability of the priority weights obtained (Tian et al. 2019). This section develops a modeling method to derive the approximate priority weights of designed schemes from PHIFPRs based on the approximate consistency measure matrix.
Similar to the above definition of consistency, the following definition of the approximate consistency measure matrix EQ for the PHIFPRQ is developed.

Remark 1
If eq i j = 0.5 + n−1 2 (w i − w j ), then eq ji = 0.5 + n−1 Proof The proof of Remark 1 is proved in "Appendix." Remark 1 shows that if the preference relationship in the upper triangular matrix is approximately consistent, then the preference relationship in the lower triangular matrix is also approximately consistent. Therefore, it is only necessary to test the approximate consistency in the upper triangular matrix of PHIFPR.
In addition, Eq. (7) is true only for an approximately consistent PHIFPR, which is unrealistic for various reasons. In this case, Eq. (7) does not hold, i.e., there are some deviations between the two sides of Eq. (7). Generally, the higher the approximate consistency index of the PHIFPR, the more rational its preference information. Consequently, the approximate priority weights can be obtained by minimizing the absolute deviations between its two sides based on the following programming model: We discover that Eq. (8) is a multi-objective programming model, from which we derive the following remarks.
Remark 2 For Eq. (8), we have Remark 3 For Eq. (8), if i = j, then we have Based on Remarks 2 and 3, Model 5 can be transformed to the following model, from which the approximate priority weights in a PHIFPR can be obtained: where d + i j and d − i j are the positive and negative deviations relative to the target of the goal ς i j , respectively. b i j and c i j are the weights corresponding to d + i j and d − i j , respectively. Without loss of generality, we suppose that all goals ς i j (i, j = 1, 2, . . . , n; j > i) are fair, i.e., b i j = c i j = 1. Consequently, Eq. (9) can be written as 1, 2, . . . , n).
The above model can be deemed as the selection process on the basis of the approximate consistency measure matrix, i.e., to choose the optimal constellation satellite designed scheme through the approximate priority weight of every designed scheme.

An effectiveness evaluation method of a constellation satellite communication system with PHIFPRs
For an effectiveness evaluation problem of a constellation satellite communication system, suppose there is a set of constellation satellite designed schemes D = {D 1 , D 2 , . . . , D n }, and e = {e 1 , e 2 , . . . , e k } is a set of evaluators. Suppose ω = {ω 1 , ω 2 , . . . , ω k } is the weight vector of evaluators, such that 0 ≤ ω m1 ≤ 1, k m1=1 ω m1 = 1. Each evaluator e m1 provides a PHIFPR,Q m1 = (q m1 i j ) n×n , wherẽ q m1 2, . . . , k). The corresponding approximate consistency measure matrices E m1 Q = (eq m1 i j ) n×n can be obtained according to Definition 14. We introduce a method of effectiveness evaluation using PHIFPRs, which follows and is described in Fig. 1.
Step 3: Check approximate consistency and group consensus. The former can be tested through Eq. (5), . , e k } through Eq. (6). If the approximate consistency level of all individual PHIFPRs is higher than the threshold CI 0 , and the group consensus of {e 1 , e 2 , . . . , e k } also meets the threshold CON 0 , then go to step 5. Otherwise, go to step 4.
Step 4: Deduce a set of modified PHIFPRs. The preference relationship should be modified by Model 4 if any PHIFPR fails to reach the acceptable consistency level or the group consensus does not meet the threshold value. Modified PHIFPRs can then be obtained with acceptable approximate consistency and group consensus.
Step 5: Calculate approximate priority weights of schemes based on the evaluation opinions of every evaluator, calculated by Model 7.
Step 7: Rank schemes according to values calculated by Eq. (10). The scheme with the largest value is best.

A numerical example
Constellation scheme design is the primary concern of constellation satellite communication system construction, which largely determines a system's complexity, cost, and effectiveness. It has been much studied, and a variety of schemes have been proposed. Therefore, it is pivotal to design an evaluation method to select the optimal scheme from a finite set of designed schemes according to multiple influential criteria, with the aim of maximizing global real-time data communication and maximizing integrated information services for users. In this section, we provide an application example to demonstrate the feasibility and effectiveness of the designed method providing methodological support to constellation satellite communication system effectiveness evaluation.
There are three types of communication satellite coverage areas: hotspot, basic, and global, each with different coverage requirements. For example, a hotspot service area requires uninterrupted coverage. The constellation can have certain coverage interruptions, usually less than a minute, for a basic service area. A global service area requires the constellation to cover the whole world. According to coverage requirements, four constellation satellite designed schemes D i (i = 1, 2, 3, 4) are selected as alternatives for effectiveness evaluation in terms of coverage, quality of service, adaptability, interrupt performance, and protection performance. In the effectiveness evaluation, the relevant departments conducted a preliminary analysis of the basic problems of constellation satellite communication system construction and promptly invited four evaluators of experienced chief e 1 , geological expert e 2 , telecommunications expert e 3 and user of constellation satellite communication system e 4 to analyze and select the best constellation satellite designed scheme. First, we produce a corresponding questionnaire based on the fundamental data of the constellation satellite designed schemes and five criteria about coverage, quality of service, adaptability, interrupt performance, and protection performance. When filling in the questionnaire, four evaluators e i (i = 1, 2, 3, 4) are suggested using PHIFPR to express their individual preferences by pairwise comparisons of four schemes. Then we can obtain four PHIFPRsQ 1 ,Q 2 ,Q 3 ,Q 4 , which are composed of PHIFNs. The weights of the evaluators are ω = (0.2, 0.3, 0.3, 0.2). The procedures for obtaining the optimal designed scheme by using the proposed method are described as follows, Step 1: Identify PHIFPRs, the individual threshold CI 0 = 0.9, and the consensus threshold CON 0 = 0.7.
Step 2: Calculate approximate consistency measure matrices. Step 3: Check approximate individual consistency and group consensus.
Step 4: Deduce a set of modified PHIFPRs. We use the GA to obtain modified preference relationshipŝ Q m1 (m1 = 1, 2, 3, 4) so that the modified PHIFPRs retain as much original information as possible while meeting the constraints. Figure 2 shows the fitness function values based on 500 iterations in GA, from which we see that the fitness function is slowly minimized, with optimum value 0.0038. The modified preference relationshipsQ m1 (m1 = 1, 2, 3, 4) can be obtained, as shown in Tables 5, 6, 7, and 8, respectively, and the approximate consistency measure matrices E m1 Q (m1 = 1, 2, 3, 4) are as follows: relationships are of acceptable approximate consistency, and Q = {Q 1 ,Q 2 ,Q 3 ,Q 4 } is the acceptable group consensus.
Step 5: Calculate approximate priority weights of schemes for every evaluator. From Model 7, we can obtain the approximate priority weights of the constellation satellite designed schemes: Step 6: Calculate overall approximate priority weights of designed schemes. From Eq. (10), we obtain: Step 7: Rank schemes. Given w 1 > w 4 > w 3 > w 2 , i.e., D 1 > D 4 > D 3 > D 2 , the optimal designed scheme is D 1 . In the evaluation, we consider the two critical problems of group consensus and individual consistency, in which group consensus aims to help evaluators achieve agreement regarding the solution to a common evaluation problem, and individual consistency checking aims to ensure the rationality of evaluation information to avoid misleading priority weights of constellation satellite design schemes. In addition, to ensure the credibility of the obtained evaluation results, the fitting error of the approximate consistency measure matrix E m1 Q (m1 = 1, 2, 3, 4) is calculated by solving model 7, we can get P 1 = 0.0490, P 2 = 0.2892, P 3 = 0.1372, 1, 2, 3, 4). As we all know, the higher the consistency of the PHIFPR, the more rational the preference information in that PHIFPR is, and the more reliable the priority weight vector of that PHIFPR is. Since the value of P m1 is relatively low in the case of retaining as much original information as possible, the ranking results obtained in this paper are of some reference value, which can provide the right direction for the selection of designed schemes.
To validate the effectiveness of the designed method, we compare it to other methods considering the same example. The method of Zhou and Xu (2018) first uses the expected consistency to improve the individual consistency level by iterative optimization based on the PHFPR during the evaluation. Then they employ general operational laws to aggregate the evaluation information of every evaluator and sort the schemes. By their method, we can obtain a final ranking of D 2 > D 4 > D 1 > D 3 , which is quite different from that obtained by our method. The reasons lead to this difference may contribute to two points: (1) the proposed method takes into account the max non-priority intensity, which represents the worst negative information and the maximum risk preference of evaluators during the effectiveness evaluation, which is more suitable and reasonable to describe psychological states of evaluators than the PHFPR in Zhou and Xu (2018), (2) the group consensus is not considered in Zhou and Xu (2018), which will result in lower satisfaction with the obtained result.  To further confirm the importance of the worst negative information and the risk preference of evaluators in the evaluation process, we employ the method of Zhou et al. (2015) based on the hesitant-intuitionistic fuzzy information with the same example. They propose an approximate consistency test to judge consistency level. Compared to the method of Zhou and Xu (2018), they consider the worst negative information and the risk preference in the evaluation. Applying this methodology to address our example, we have D 1 > D 4 > D 2 > D 3 , which implies that the best designed scheme proposed by the method in Zhou et al. (2015) is the same as ours, but the ranking orders of constellation satellite designed schemes are different. The reasons lead to this difference may contribute to two points: (1) the method in Zhou et al. (2015) also ignores group consensus problem, resulting in evaluators not achieving agreement regarding the solution to a common evaluation problem, (2) the method in Zhou et al. (2015) ignore the occurrence probability of each evaluation value, making it impossible for evaluators to accurately express their individual opinions. Table 9 provides a comparative analysis of three evaluation methods. From Table 9, we can find that literature Zhou and Xu (2018) and Zhou et al. (2015) both ignore group consensus in the evaluation process. The evaluation process involves many evaluators with different interests, educational backgrounds, social statuses, and different understandings of the evaluation problem, leading to conflicts. Thus, the consensus reaching process is necessary for the effectiveness evaluation. In addition, priority weight can be used to generate the ranking order of constellation satellite designed schemes, but the individual preferences provided by evaluators are usually inconsistent, and the priority weights obtained from the inconsistency matrix cannot guarantee the reliability of the evaluation results. Therefore, it is very important to determine the priority weights from the preference relations in the evaluation process, but the literature Zhou et al. (2015) failed to consider this problem. Given the above analysis, the method proposed in this paper can be more practical and reasonable compared to the existing methods.
In conclusion, our method can minimize the controversy caused by the optimal result and can more accurately describe the evaluation information of each evaluator than other methods. The main innovations of the proposed method are included as follows.
-We define a new preference relationship, PHIFPR, which can describe the hesitance of evaluators and the hesitance distribution information in the process of evaluation, consider the worst negative information, and relate the risk preference in the form of a non-membership degree. Therefore, PHIFPR can more appropriately describe evaluators' individual views, thereby avoiding the loss modified PHIFPRs based on minimal information adjustment, by which the obtained judgment matrices have a higher individual consistency and group consensus. Furthermore, the model retains the original evaluation information as much as possible. -The approximate priority weights of schemes can be objectively determined using the approximate consistency measure matrix of PHIFPRs. Generally, the higher the approximate consistency of a PHIFPR, the more rational preference information is and the more reliable its approximate priority weights.
Nevertheless, the proposed method has limitations. It is difficult to determine the threshold values of CI 0 and CON 0 . To date, there has been no unified approach to set CI 0 and CON 0 . The determination of the consistency or consensus threshold values depends on the subjective preferences of evaluators and the actual evaluation environment. When the evaluation is urgent, smaller threshold values could be considered suitable; otherwise, higher threshold values should be chosen to determine a solution more suited to the situation.

Conclusions
Individual consistency and group consensus are significant research topics in effectiveness evaluation and are of increasing concern. This paper introduces an effectiveness evaluation method for a constellation satellite communication system by examining and improving the approximate consistency and group consensus with PHIFPRs. The approximate consistency index for the PHIFPR is first deduced based on the proposed PHIFPR and designed approximate consistency measure matrix, and a programming model is established to improve the consistency of PHIFPRs when necessary. A group consensus index among evaluators is presented, and a programming model is introduced to simultaneously achieve the desired approximate consistency and group consensus. What's more, we propose a programming model based on the approximate consistency measure matrix to calculate the approximate priority weights of each constellation satellite designed scheme, from which a suitable optimal designed scheme can be obtained. Finally, we present some comparative analyses to illustrate the merits of the developed method through a case study. It is shown that our effectiveness evalu-ation method is more flexible and practical than the existing evaluation methods, and effective at evaluating and ranking constellation satellite designed schemes under an uncertain context.
Given the aforementioned conclusions, it is found that the whole evaluation process based on PHIFPR is given. As researchers point out, it is difficult to ask evaluators to give numerical judgments due to the complexity of the evaluation problem. Given that, this paper introduces a more general form of expressing preference called PHIFPR, where evaluators draw on PHIFPR to describe their individual preferences endowing evaluators with more rights to denote their qualitative judgments. In addition, to ensure the credibility of the obtained evaluation results, an approximate consistency checking method suitable for PHIFPR is designed, which can be used to determine the reliable ranking orders of constellation satellite designed schemes. The proposed method provides a basis for the comparative selection of schemes and provides technical support for developing satellite communication technology.
The proposed effectiveness evaluation method also has some drawbacks that suggest directions for future research.
-Evaluators' weights are given in advance in the proposed method, which is excessively dependent on evaluators' knowledge, and may be dangerous and open to manipulation. Therefore, objective weighting methods should be developed to dynamically determine the weights of evaluators, depending on the environment and accumulated knowledge (Pang et al. 2017;). -We only considered comprehensive evaluation information based on our proposed evaluation criteria. To consider evaluation information under different criteria can provide more accurate and effective results. We wish to carry out an effectiveness evaluation of constellation satellite designed schemes under different criteria, and at the same time to analyze the interrelationships among multiple evaluation criteria. -We consider the consensus problem of small group evaluation in this paper, but in practical evaluation, large-scale group evaluation (LSGE) is more worthy of attention as it can bring together more expert knowledge to help obtain better evaluation results. Therefore, in future work, we will carry out the evaluation of constellation satellite communication system based on LSGE.
This completes the proof of Theorem 1.

Proof of Remark 1
LetQ = (q i j ) n×n be a PHIFPR, whereq i j = r i j ( p i j ), ν i j with r i j ( p i j ) = {r l i j ( p l i j )|l = 1, 2, . . . , L r i j }, and ν i j = 1 − max{r i j }. According to Definitions 8 and 14, when L r i j =1, we can easily get eq ji = 0.5 + n−1 2 (w j − w i ), so we will not elaborate on it here. When L r i j > 1, 1) if 0 < S(q ji ) + (1 − r l i j ) p l i j − min{r i j } + max{r i j } − max{r i j } − 1 + 1 + 1 − max{r ji } + min{r ji } 2 = 1 + 1 − Lr i j l=1 r l i j p l i j − max{r i j } + 1 − max{r ji }+2 · max{r i j }+ min{r ji }−2 · min{r i j } − 2 2 = 1 − eq i j = 0.5 + n − 1 2 (w j − w i ).
2) if S(r ji ) + (1 − r l i j ) p l i j − min{r i j } + max{r i j } − max{r i j } − 1 + 1 + 1 − max{r ji } + min{r ji } 2 − ⎛ ⎝ Lr i j l=1 (1 − r l i j ) p l i j − min{r i j } + max{r i j } − max{r i j } − 1 + 1 + 1 − max{r ji } + min{r ji } 2 where a i j = S(q i j ) + 1−max{r i j }+min{r i j } 2 , and a i j denotes the smallest integer greater than a i j .
The proof of Remark 1 is completed.