An improved method for satellite emergency mission scheduling scheme group decision-making incorporating PSO and MULTIMOORA

Abstract

Satellite emergency mission scheduling scheme group decision making (SEMSSGDM) is a key part of satellite mission scheduling research. An appropriate evaluation model can provide a dependable and sustainable improvement and guide the functioning of emergency mission scheduling. Consequently, this research is devoted to proposing a novel decision-making method that employs a novel consensus model with hesitant fuzzy 2-tuple linguistic sets (HF2TLSs) to eliminate disagreements among satellite dispatchers and reach consensus in scheme decision-making. Within the novel method, it proposes a distance measurement function based on Hausdorff distance with HF2TLS to gauge the fit and similarity across satellite dispatchers. Additionally, a consensus reaching process (CRP) is designed to adjust the judgement of satellite dispatchers taking into account the trust degree to improve consensus. Within the selection process, a combination of the particle swarm optimization (PSO) algorithm and the MULTIplicative MOORA (MULTIMOORA) method is applied, where PSO is performed to improve the accuracy of information aggregation, and the MULTIMOORA method is used to develop the robustness of the selection results. Lastly, an applicative example validates the effectiveness of the method based on a mission scheduling intelligent decision simulation system.

Keywords

Satellite emergency mission scheduling scheme group decision-making hesitant fuzzy 2-tuple linguistic set consensus reaching processes(CRPs)particle swarm optimization(PSO)MULTIplicative MOORA (MULTIMOORA)

Abbreviation

SEMSSGDM

satellite emergency mission scheduling scheme group decision making

HF2TLSs

hesitant fuzzy 2-tuple linguistic sets

CRPs

consensus reaching processes

PSO

particle swarm optimization

MOORA

multiobjective optimization by ratio analysis

MULTIMOORA

Multiplicative MOORA

DMs

decision makers

GDM

group decision-making

FMF

full-multiplicative form

ratio system

1 Introduction

Earth observation satellites have the advantages of wide observation range, permanent or continuous observation and not restricted by national boundaries and geographical conditions. With the frequently occurrence of emergency events such as earthquakes, water and flood disasters and volcanic eruptions, the using of satellites for reconnaissance and observation has been increasingly valued by national disaster prevention and mitigation departments. Such emergencies are usually high-dynamic and time-sensitive, requiring rapid scheduling and generating multiple sets of emergency mission scheduling schemes. On this footing, determining an overall optimal emergency scheme for observation is a critical issue, and the decision-making demand emerges. Its primary objective is to select the best observation scheme to achieve maximum satisfaction of remote sensing image needs for different users, different requirement grades and different target types within a limited time frame. Scientific and rational decision-making on satellite emergency mission scheduling schemes applied to emergency services can provide some feedback to dispatchers for adjusting and optimising the weaknesses of the schemes, and provide support for subsequent emergency observation strategy optimisation and system optimisation design.

Accompanying complexity and uncertainty, an satellite emergency mission scheduling scheme decision-making process requires the involvement of multiple decision makers (DMs), which can lead to the leading to the can lead to the group decision-making (GDM) problem [1]. Such satellite emergency mission scheduling scheme group decision-making (SEMSSGDM) problems are characterized by the following four features [2]: (a) DMs are usually involved from different positions, (b) eventual decisions have to be made within a relatively short time, (c) DMs frequently find it difficult to reach a consensus at once, and (d) a wrong decision or one that is too slow can lead to catastrophic losses. A SEMSSGDM problem can consequently be simplified to the consensus processes by which a set of DMs with different positions quickly and accurately select the best alternative from a limited set of emergency alternatives. Generally, there are three phases in the process of GDM [3]: the preference information phase; the consensus phase; and the selection phase.In the preference information phase, DMs can provide various preference formats when evaluating their opinions on alternatives, such as intuitionistic fuzzy preference relations [4], fuzzy linguistic preference relations [5], probabilistic hesitant fuzzy preference relations [6, 7], interval-valued preference relations [8], etc. Considering that once the DM gives his decision, the next significant concern is how to aggregate individual opinions and reach consensus [3]. The selection phase comprises two different steps: aggregation of individual preference relations and exploitation of collective preference relations [3].

In the preference information phase, because of uncertainties in mission scheduling, ambiguities in the dispatcher’s thought, and the complexity of the actual satellite scheme decision-making problem, the decision information cannot be expressed in precise numbers and can only provide uncertain opinions [9]. DMs often make judgments with varying degrees of hesitation or exhibit some degree of lack of knowledge, making it difficult to express cognitive outcomes in a definitive linguistic terminology, and definitive linguistic variables are challenged in the application of decision problems [10]. Consequently, Herrera and Martinez [11] first proposed to describe linguistic evaluation information with 2-tuple linguistics. Concurrently, to avoid the information missing in the evaluation process and to promote the accuracy of the decision results, Rodriguez [12] defined the hesitant fuzzy linguistic term set (HFLTS) in accordance with the ideology of hesitant fuzzy sets [13]. Hereby, decision makers can evaluate a linguistic variable with more flexible linguistic expressions. At present, there are many researches on HFLTS [14, 15]. HFLTS supports decision making by linguistic description under qualitative environments. HFLTS can tackle situations in which DMs consider multiple potential linguistic terms at the same time than a single term for an indicator, alternative, variable, etc., to express their preferences without the use of numerical values [16]. Krishankumar et al. [17] presented a new extension of HFLTS called intuitionistic fuzzy confidence-based hesitant fuzzy linguistic term set [18]. They also adopted a double hierarchy hesitant fuzzy linguistic term set to convey complex linguistic information [19]. Subsequently, considering the respective advantages of HFLTS and 2-tuple linguistic term, Rodriguez et al. [20] further proposed the hesitant fuzzy 2-tuple linguistic set (HF2TLS) to realize more precise descriptions of linguistic preferences and abolishes the claim of linguistic scalar certitude therein. Based on this, this paper solves the SEMSSGDM problem with the characteristics of HF2TLS to alleviate the inaccuracy of evaluation information.

In the consensus phase, consensus reaching processes (CRPs) [21] attempt to remove the disagreement among dispatchers to obtain an agreed scheme. In the SEMSSGDM, these dispatchers come from different positions such as chiefs, intelligence officers. In CRPs, the study of consensus modeling has received increasing attention in the field of decision science [22, 23]. In general, improving consensus involves mathematical modeling or iterative algorithms, with the former saving time and being more applicable in SEMSSGDM. Distance measures are the basis for dealing with group consensus. Xu [24] defined the concepts of deviation degree and similarity degree between two linguistic values, which laid the theoretical foundation for the application of linguistic distance measures in GDM. Currently, some distance measures such as hesitant Hamming distance and hesitant Euclidean distance can be directly applied to the distance calculation between HFLTSs, which can disregard the number of elements in the set [25 –28]. Hausdorff distance is a measure of how similar two nonempty sets in a space are to each other in terms of position, and is mostly applied in remote sensing image matching to determine the distance between each model point and the nearest image point [29]. Compared with the traditional Hamming distance and Euclidean distance measures, Hausdorff distance can calculate the distance between two sets faster and more accurately. Consequently, this paper attempts to replace the distance measure formula in the traditional consensus model with the Hausdorff distance, and proposes a Hausdorff distance measure function directly for HF2TLSs based on following the classical Hausdorff distance, on which to achieve the CRPs.

In the selection phase, studies have been conducted on satellite scheduling algorithms, satellite effectiveness, and the GDM process of satellite solutions based on decision-making methods such as analytical hierarchy process (AHP) [30, 31], the technique for order preference by similarity to an ideal solution (TOPSIS) [32], fuzzy theory (FCE) [33], and ADC model [34], etc. Within the SEMSSGDM process, the aggregation accuracy of individual decision-making information and the robustness of the scheme selection results deserve considerable attention. The reliability of the SEMSSGDM results affects the decision-making of the whole satellite mission scheduling system, consequently it requires a combination of methods to improve the decision-making accuracy. PSO is a global optimization evolutionary algorithm proposed by Kennedy and Eberhart [35]; this algorithm is the most widely used and efficient evolutionary algorithm [36]. PSO has a wide range of advantages over genetic algorithms (GA). For example, the PSO algorithm helps to optimize the solution at each stage and its characteristics facilitate fast convergence to the solution, whereas other metaheuristic algorithms, such as GA, do not have this bootstrap mechanism. Moreover, in PSO, it doesn’t requires any external parameters to tune the algorithm, while GA requires some parameters such as crossover, variance, etc. [37], which makes PSO suitable for SEMSSGDM problems. Applying PSO to this study, the aggregation goes through the individual decision-making matrix of CRPs, so the particles in PSO are one by one matrix, and after the evolutionary calculation of PSO, we should finally get an aggregated population matrix, and the matrix is the scheme-attribute matrix. Also, it is analyzed that in our everyday life, we experience various situations, where we need a mathematical function having the ability to reduce a set of numbers into a unique representative one [37]. To obtain the final scheme ranking, many scholars have studied many methods [30 , 33]. The problem with the above methods is that only one ranking occurs, which can create uncertainty and instability. The multiobjective optimization by ratio analysis(MOORA) method was originally introduced by Brauers and Zavadskas [38] and further enhanced by adding the full-multiplicative form (FMF) to generate the MULTIplicative MOORA (MULTIMOORA) method [39]. It consists of three aggregation models with different functions: the RS method, the RP model, and FMF procedure. In comparison with AHP, TOPSIS, VIKOR, PROMETHEE, LINMAP, and ELECTRE, the MULTIMOORA approach has more superiority, easy mathematical expressions, less computation time, and strong robustness [40]. Due to its unique advantages over other GDM methods, the classical MULTIMOORA method has been employed for various GDM concerns [41, 42]. Therefore, based on the fast convergence of PSO, the MULTIMOORA method combines the features of the three GDM methods and is an effective way to solve realistic emergency problems and improve the robustness of the results.In summary, the main goal of this paper is to develop an improved method for satellite emergency mission scheduling scheme group decision-making incorporating PSO and MULTIMOORA. The main innovation points and contributions are presented below.

Designing novel Hausdorff distance functions to accurately measure the distance between HF2TLSs The Hausdorff distance is a well-established measure for computing the distance between two sets. The Hausdorff distance based on the HF2TLS extension can determine the distance between any two HF2TLSs with different number of elements.

Building a new consensus model. The model takes into account the degree of trust between dispatchers when determining their weights, and the decision-making matrix adjustment of non-consensus members is done under the guidance of other members.

For improving the accuracy of SEMSSGDM results, combining PSO and MULTIMOORA methods to optimise the scheme selection process. First, individual decision-making information is aggregated based on PSO in the aggregation process, and then enhancing the robustness of SEMSSGDM results based on the MULTIMOORA method in the scheme ranking process.

The rest of this paper is organized as follows. Section 2 briefly reviews some fundamental concepts of hesitant fuzzy 2-tuple linguistic set (HF2TLS), and Hausdorff distance. Section 3 proposes the Hausdorff distance function with HF2TLSs and builds a consensus model based on it for designing CRPs. Section 4 proposes a method to solve the SEMSSGDM problem as per HF2TLS and presents the steps to combine the PSO method and MLTIMOORA method to aggregate the decision-making information and optimize the decision-making results in the scheme selection process. In Section 5, an application example using the proposed method to the SEMSSGDM problem is given, to be followed by a comparative analysis. Finally, conclusions are drawn in Section 6.

2 Preliminaries

In this section, we briefly review several basic definitions regarding HF2TLS and the concept of the Hausdorff distance.

2.1 Hesitant fuzzy 2-tuple linguistic set

HF2TLS integrates the merits of 2-tuple linguistic and HFLTS. Its description is more accurate and can avoid information loss during information processing and decision computation. Compared with HFLTS, all possible values of the hesitant fuzzy linguistic set can be directly considered. Thereby, the decision information contained in HF2TLS is more comprehensive.

Definition 1. [43] Let S = {s₀, s₁, …, s_g-1, s_g} be a linguistic term set, and β ∈ [0, g] be a value representing the result of a symbolic aggregate operation. Then, the 2-tuple that expresses the equivalent information to β is obtained by the following function: $\begin{matrix} △ : [0, g] \to S \times [- 0.5, 0.5), \\ △ (β) = (s_{ι}, α), with \\ {\begin{matrix} s_{ι}, & i = round (β), \\ α = β - i . & α \in [- 0.5, 0.5) . \end{matrix}, \end{matrix}$ where round (.) is a rounded function that returns a positive number β. On the contrary, the 2-tuple linguistic (s_ι, α_ι) can be reduced to the equivalent value β ∈ [0, g] through the function Δ^-1: $\begin{matrix} Δ^{- 1} : S \times [- 0.5, 0.5) \to [0, g], \\ Δ^{- 1} (s_{ι}, α_{ι}) = i + α = β . \end{matrix}$

Definition 2. [43] Suppose (s_ι, α_ι) and (s_j, α_j) are two 2-tuple linguistic sets defined on the linguistic term sets S = {s₀, s₁, …, s_g-1, s_g}, and the comparison rules are

$\begin{array}{l} (1) If i < j, then (s_{ι}, α_{ι}) < (s_{j}, α_{j}); \\ (2) If i = j a n d \\ \begin{matrix} ➀ α_{ι} = α_{j}, then (s_{ι}, α_{ι}) = (s_{j}, α_{j}); \\ ➁ α_{ι} < α_{j}, then (s_{ι}, α_{ι}) < (s_{j}, α_{j}); \\ ➂ α_{ι} > α_{j}, then (s_{ι}, α_{ι}) > (s_{j}, α_{j}) . \end{matrix} \end{array}$

Definition 3. [44] Let S = {s₀, s₁, …, s_g} be a linguistic term set. If H_s is a set of a finite number of sequential consecutive linguistic terms in S, then H_s is called a hesitant fuzzy linguistic term set on S, and is abbreviated as HFLTS. In order to avoid the loss of information in the decision-making process and improve the accuracy of the decision-making results, the concept of hesitant fuzzy 2-tuple linguistic set is introduced below.

Definition 4. [44] Let S = {s₀, s₁, …, s_g} be a linguistic term set, (s_ι, α_ι) is the 2-tuple linguistic set on S, satisfying ∀i < j, (s_ι, α_ι) < (s_j, α_j). Then H_T = {(s_ι, α_ι) |i = 1, 2, …, l (H_T)} is called a hesitant fuzzy 2-tuple linguistic set on S, and is abbreviated as HF2TLS. l (H_T) is the granularity of H_T.

Example 1. In order to evaluate the scheme performance of the four emergency alternatives, a set of linguistic terms is given in advance and it is assumed that the decision maker considers the scheme performance of the four options to be "between poor and fair, between very good and exceptionally good, very good, good". For each of these linguistic terms, the decision maker has his or her own hesitant value for the sign transfer value. Thus, the scheme performance of each of the four alternatives is expressed utilizing HF2TLSs as { (s₂, 0.2) , (s₃, 0.1)}, { (s₅, 0.25) , (s₆, 0.3)}, { (s₅, 0.35)}, { (s₄, 0)}. Taking advantage of the HF2TLSs can be denoted as { (x₁, { (s₂, 0.2) , (s₃, 0.1) }) , (x₂, { (s₅, 0.25) , (s₆, 0.3) }) , (x₃, { (s₅, 0.35) }) , (x₄, { (s₄, 0) }) }

Definition 5. [45] Let H_T = {(s_ι, α_ι) |i = 1, 2, …, l (H_T)}, and it is defined as a hesitant fuzzy 2-tuple linguistic set on S = {s₀, s₁, …, s_g}. l (H_T) is the granularity of H_T. Then, the expected function of H_T is

$E (H_{T}) = \frac{1}{l (H_{T})} \sum_{i = 1}^{l (H_{T})} Δ^{- 1} (s_{ι}, α_{ι}) .$ (1)

2.2 The Hausdorff distance

The Hausdorff distance is a measure of how much two nonempty sets A and B in a metric space resemble each other with respect to their positions [28]. It is the “maximum distance of a set to the nearest point in the other set” [46, 47]. More formally, the Hausdorff distance from A to B is calculated by performing a max–min operation on the elements belonging to the corresponding set: $\begin{matrix} h (A, B) & = & max_{a \in A} {min_{b \in B} {d (a, b)}}, \end{matrix}$ where d (a, b) denotes any distance between a and b. In addition, h (A, B) is usually not equal to h (B, A), that is to say, the Hausdorff distance is directional. A more general Hausdorff distance is represented as $\begin{matrix} H (A, B) = max {h (A, B), h (B, A)} . \end{matrix}$ This function defines the Hausdorff distance between A and B, and h (A, B) is the Hausdorff distance from A to B (also called the directed Hausdorff distance). h (A, B) and h (B, A) are sometimes referred to as the forward and backward Hausdorff distances from A to B, respectively. In a nutshell, H (A, B) measures the difference between matching points and the image points [48]. Therefore, it helps to determine the distance between the model point and closest image point.

3 Novel consensus reaching model considering the distance and consensus degree

In this section, a consensus index function is proposed for SEMSSGDM in terms of HF2TLS over Hausdorff distance. The consensus between two HF2TLSs is finally attributed to iterating the distance between hesitant fuzzy 2-tuple linguistic sets. Therefore, a novel HF2TLS distance measurement function based on the Hausdorff distance is first designed. In one stage, the similarity between decision-making matrices of multi-attribute schemes for satellite dispatchers is calculated. Finally, a consensus index based on HF2TLS is constructed to calculate and evaluate the matrices consensus degree.

3.1 The Hausdorff distance for HF2TLSs

As described in Section 2.2, the Hausdorff distance is a deterministic method suitable for measuring the distance between two point sets. The Hausdorff distance function between HF2TLSs is defined as

Definition 6. Let $r^{1} = {(s_{ι}^{1}, α_{ι}^{1}) | i = 1, 2, \dots, l (r^{1})} and r^{2} = {(s_{ι}^{2}, α_{ι}^{2}) | i = 1, 2, \dots, l (r^{2})}$ be two HF2TLSs on S = {s₀, s₁, …, s_g}. The hesitant fuzzy 2-tuple linguistic Hausdorff distance between r¹ and r² is defined as follows: $D (r^{1}, r^{2}) = max {d (r^{1}, r^{2}), d (r^{2}, r^{1})}$ (2) $\begin{matrix} = max {max_{(s_{ι}^{1}, α_{ι}^{1}) \in r^{1}} min_{(s_{ι}^{2}, α_{ι}^{2}) \in r^{2}} | \frac{(s_{ι}^{1} + α_{ι}^{1}) - (s_{ι}^{2} + α_{ι}^{2})}{g} |, \\ max_{(s_{ι}^{2}, α_{ι}^{2}) \in r^{2}} min_{(s_{ι}^{1}, α_{ι}^{1}) \in r^{1}} | \frac{(s_{ι}^{2} + α_{ι}^{2}) - (s_{ι}^{1} + α_{ι}^{1})}{g} |}, \end{matrix}$

where d (r¹, r²) is the Hausdorff distance from r¹ to r².

The proposed hesitant fuzzy linguistic Hausdorff distance is calculated directly using HF2TLSs and there is no need to add any value into the set with a fewer number of elements.

Property 1. Let R be the set of all HF2TLSs. The hesitant fuzzy 2-tuple linguistic Hausdorff distance satisfies the following properties:

0≤ D (r¹, r²) ≤1, ∀ r¹, r² ∈ R ;

D (r¹, r¹) =0, ∀ r¹ ∈ R ;

D (r¹, r²) = D (r², r¹) , ∀ r¹, r² ∈ R ;

∀r¹, r², r³ ∈ R, if r¹ ≤ r² ≤ r³, then D (r¹, r³) ≥ D (r¹, r²) and D (r¹, r³) ≥ D (r², r³) .

Proof. The proof of properties 2 and 3 is obvious, the proof is omitted in this paper.

1. By Definition 2.1, knowing that $(s_{ι}^{1} + α_{ι}^{1}) \in [0, g], (s_{ι}^{2} + α_{ι}^{2}) \in [0, g] . Since, for r^{1}, r^{2} \in R, 0 \leq \frac{s_{ι}^{1} + α_{ι}^{1}}{g} \leq 1, 0 \leq \frac{s_{ι}^{2} + α_{ι}^{2}}{g} \leq 1 . Hence, 0 \leq | \frac{(s_{ι}^{1} + α_{ι}^{1}) - (s_{ι}^{2} + α_{ι}^{2})}{g} | \leq 1, 0 \leq d (r^{1}, r^{2}) \leq 1 . Similarly, 0 \leq d (r^{2}, r^{1}) \leq 1 . Finally, 0 \leq D (r^{1}, r^{2}) \leq 1 .$

4. For any there HF2TLSs, if $r^{1}, r^{2}, r^{3} \in R, therefore, 0 \leq \frac{s_{ι}^{1} + α_{ι}^{1}}{g} \leq \frac{s_{ι}^{2} + α_{ι}^{2}}{g} \leq \frac{s_{ι}^{3} + α_{ι}^{3}}{g} \leq 1 . Thus, \frac{s_{ι}^{3} + α_{ι}^{3}}{g} - \frac{s_{ι}^{1} + α_{ι}^{1}}{g} \geq \frac{s_{ι}^{2} + α_{ι}^{2}}{g} - \frac{s_{ι}^{1} + α_{ι}^{1}}{g} \geq 0 \Rightarrow | \frac{s_{ι}^{1} + α_{ι}^{1}}{g} - \frac{s_{ι}^{3} + α_{ι}^{3}}{g} | \geq | \frac{s_{ι}^{1} + α_{ι}^{1}}{g} - \frac{s_{ι}^{2} + α_{ι}^{2}}{g} | \Rightarrow min_{(s_{ι}^{3}, α_{ι}^{3}) \in r^{3}} | \frac{(s_{ι}^{1} + α_{ι}^{1}) - (s_{ι}^{3} + α_{ι}^{3})}{g} | \geq min_{(s_{ι}^{2}, α_{ι}^{2}) \in r^{2}} | \frac{(s_{ι}^{1} + α_{ι}^{1}) - (s_{ι}^{2} + α_{ι}^{2})}{g} | \Rightarrow max_{(s_{ι}^{1}, α_{ι}^{1}) \in r^{1}} min_{(s_{ι}^{3}, α_{ι}^{3}) \in r^{3}} | \frac{(s_{ι}^{1} + α_{ι}^{1}) - (s_{ι}^{3} + α_{ι}^{3})}{g} | \geq max_{(s_{ι}^{1}, α_{ι}^{1}) \in r^{1}} min_{(s_{ι}^{2}, α_{ι}^{2}) \in r^{2}} | \frac{(s_{ι}^{1} + α_{ι}^{1}) - (s_{ι}^{2} + α_{ι}^{2})}{g} | . Similarly, max_{(s_{ι}^{3}, α_{ι}^{3}) \in r^{3}} min_{(s_{ι}^{1}, α_{ι}^{1}) \in r^{1}} | \frac{(s_{ι}^{3} + α_{ι}^{3}) - (s_{ι}^{1} + α_{ι}^{1})}{g} | \geq max_{(s_{ι}^{2}, α_{ι}^{2}) \in r^{2}} min_{(s_{ι}^{1}, α_{ι}^{1}) \in r^{1}} | \frac{(s_{ι}^{2} + α_{ι}^{2}) - (s_{ι}^{1} + α_{ι}^{1})}{g} |$ .

Therefore, d (r¹, r³) ≤ d (r¹, r²) and d (r³, r¹) ≥ d (r², r¹) . SinceD (r¹, r³) = max { d (r¹, r³) , d (r³, r¹) } , and D (r¹, r²) = max { d (r¹, r²) , d (r², r¹) } , then D (r¹, r³) ≥ D (r¹, r²) isobtained . Analogously, D (r¹, r³) ≥ D (r², r³) can also be proved.

In order to measure the distance between two decision-making matrices of HF2TLSs, this paper defines the distance function based on decision-making matrices [22].

Definition 7. Let $R_{S}^{p} = {(r_{ij}^{p})_{m \times n}, r_{ij}^{p} = {(s_{ι}^{p}, α_{ι}^{p}) | ι = 1, 2, \dots, l (r_{ij}^{p})}}$ and $R_{S}^{q} = {(r_{ij}^{q})_{m \times n}, r_{ij}^{q} = {(s_{ι}^{q}, α_{ι}^{q}) | ι = 1, 2, \dots, l (r_{ij}^{q})}}$ be any two decision-making matrices based on any two satellite dispatchers, namely, the decision-making information of m satellite schemes to the n decision-making attributes of satellite schemes. $r_{ij}^{p}$ and $r_{ij}^{q}$ is composed of several HF2TLSs, the number of which is determined by the dispatcher. The distance between two decision-making matrices can be denoted as follows:

$D_{e} (R_{S}^{p}, R_{S}^{q}) = \frac{1}{m \times n} \sum_{i = 1}^{m} \sum_{j = 1}^{n} D (r_{ij}^{p}, r_{ij}^{q}) .$ (3)

The consensus index of HF2TLSs can be obtained by utilizing the above distance function. This is discussed in the following subsection.

3.2 Consensus index with HF2TLSs

The consensus index of the HF2TLSs is defined as follows. Calculating the similarity degree of any two dispatchers’ decision-making matrices $R_{S}^{p}$ and $R_{S}^{q}$ is the first step to obtain group consensus. Therefore, this paper suppose that the similarity degree $SIM (R_{S}^{p}, R_{S}^{q})$ is determined by the Hausdorff distance in Definition 3.1. On the basis of $SIM (R_{S}^{p}, R_{S}^{q})$ , the fitting degree $FIT (R_{S}^{p})$ between dispatchers $R_{S}^{p}$ and other dispatchers can be obtained. Then, the group consensus index CON (R) among ${R_{S}^{1}, R_{S}^{2}, \dots, R_{S}^{f}}$ can be acquired based on the fitting degree. $SIM (R_{S}^{p}, R_{S}^{q}) = 1 - D_{e} (R_{S}^{p}, R_{S}^{q})$ (4) $\begin{matrix} = & 1 - \frac{1}{m \times n} \sum_{i = 1}^{m} \sum_{j = 1}^{n} D (r_{ij}^{p}, r_{ij}^{q}) \\ = & 1 - \frac{1}{m \times n} \sum_{i = 1}^{m} \sum_{j = 1}^{n} [max {max_{(s_{ι}^{p}, α_{ι}^{p}) \in r_{ij}^{p}} min_{(s_{ι}^{q}, α_{ι}^{q}) \in r_{ij}^{q}} \\ | \frac{(s_{ι}^{p} + α_{ι}^{p}) - (s_{ι}^{q} + α_{ι}^{q})}{g} |, max_{(s_{ι}^{q}, α_{ι}^{q}) \in r_{ij}^{q}} min_{(s_{ι}^{p}, α_{ι}^{p}) \in r_{ij}^{p}} \\ | \frac{(s_{ι}^{q} + α_{ι}^{q}) - (s_{ι}^{p} + α_{ι}^{p})}{g} |}] . \end{matrix}$

After got the similarity degree of any two dispatchers’ decision-making matrices such as between $R_{S}^{p}$ and other satellite dispatchers’ decision-making matrices $R_{S}^{1}, R_{S}^{2}, \dots, R_{S}^{p - 1}, R_{S}^{p + 1}, \dots, R_{S}^{f}$ . Let $\sum_{q = 1, q \neq p}^{f} SIM (R_{S}^{p}, R_{S}^{q})$ be the whole similarity degree that interrelate $R_{S}^{p}$ and the remaining dispatchers. $FIT (R_{S}^{p}) (p = 1, 2, \dots, f) =$ $\sum_{q = 1, q \neq p}^{f} SIM (R_{S}^{p}, R_{S}^{q})$ can be calculated as $\begin{matrix} FIT (R_{S}^{p}) = \frac{1}{f - 1} \sum_{q = 1, q \neq p}^{f} SIM (R_{S}^{p}, R_{S}^{q}) \\ = & \frac{1}{f - 1} \sum_{q = 1, q \neq p}^{f} (1 - \frac{1}{m \times n} \sum_{i = 1}^{m} \sum_{j = 1}^{n} [max \\ {max_{(s_{ι}^{p}, α_{ι}^{p}) \in r_{ij}^{p}} min_{(s_{ι}^{q}, α_{ι}^{q}) \in r_{ij}^{q}} | \frac{(s_{ι}^{p} + α_{ι}^{p}) - (s_{ι}^{q} + α_{ι}^{q})}{g} |, \\ max_{(s_{ι}^{q}, α_{ι}^{q}) \in r_{ij}^{q}} min_{(s_{ι}^{p}, α_{ι}^{p}) \in r_{ij}^{p}} | \frac{(s_{ι}^{q} + α_{ι}^{q}) - (s_{ι}^{p} + α_{ι}^{p}}{g} |}]) . \end{matrix}$

Let $\sum_{p = 1}^{f} FIT (R_{S}^{p})$ indicate the overall fitting degree of all dispatchers. The consensus index CON (R) among all dispatchers R₁, R₂, …, R_f can be denoted as $\begin{matrix} CON (R) \\ = & \frac{1}{f (f - 1)} \sum_{p = 1}^{f - 1} \sum_{q = p + 1}^{f} SIM (R_{S}^{p}, R_{S}^{q}) \\ = & \frac{1}{f (f - 1)} \sum_{p = 1}^{f - 1} \sum_{q = p + 1}^{f} (1 - \frac{1}{m \times n} \sum_{i = 1}^{m} \sum_{j = 1}^{n} \\ [max {max_{(s_{ι}^{p}, α_{ι}^{p}) \in r_{ij}^{p}} min_{(s_{ι}^{q}, α_{ι}^{q}) \in r_{ij}^{q}} | \frac{(s_{ι}^{p} + α_{ι}^{p} - (s_{ι}^{q} + α_{ι}^{q})}{g} |, \\ max_{(s_{ι}^{q}, α_{ι}^{q}) \in r_{ij}^{q}} min_{(s_{ι}^{p}, α_{ι}^{p}) \in r_{ij}^{p}} | \frac{(s_{ι}^{q} + α_{ι}^{q}) - (s_{ι}^{p} + α_{ι}^{p}}{g} |}]) . \end{matrix}$ $S (R_{S}^{p}, R_{S}^{q}) = S (R_{S}^{q}, R_{S}^{p})$ can be simplified as follows:

$CON (R)$ (5) $\begin{matrix} = & \frac{2}{f (f - 1)} \sum_{p = 1}^{f - 1} \sum_{q = p + 1}^{f} (1 - \frac{1}{m \times n} \sum_{i = 1}^{m} \sum_{j = 1}^{n} \\ [max {max_{(s_{ι}^{p}, α_{ι}^{p}) \in r_{ij}^{p}} min_{(s_{ι}^{q}, α_{ι}^{q}) \in r_{ij}^{q}} | \frac{(s_{ι}^{p} + α_{ι}^{p} - (s_{ι}^{q} + α_{ι}^{q})}{g} |, \\ max_{(s_{ι}^{q}, α_{ι}^{q}) \in r_{ij}^{q}} min_{(s_{ι}^{p}, α_{ι}^{p}) \in r_{ij}^{p}} | \frac{(s_{ι}^{q} + α_{ι}^{q}) - (s_{ι}^{p} + α_{ι}^{p}}{g} |}]) . \end{matrix}$

It is easy to know that 0 ≤ CON (R) ≤1; moreover, the larger the CON (R), the higher the consensus dispatchers. Based on the above function, set the consensus degree threshold CON₀ to judge the group consensus degree of the satellite dispatchers for scheme decision-making. If CON (R) ≥ CON₀, there is an acceptable consensus among all of the satellite dispatchers. Otherwise, the consensus is unacceptable.

Next, establish a novel consensus adjustment model to ensure that the group consensus is increased when the consensus is unacceptable.

3.3 Improved consensus model for SEMSSGDM problem

In order to achieve a scheme to the SEMSSGDM problem which is accepted by the whole group, CRPs have been paid great attention as part of the schedule process. It is important to improve consensus before satellite dispatchers choose a final scheme; furthermore, CRPs are a dynamic and iterative process. In the previous subsection, we designed a consensus index function CON (R) based on the Hausdorff distance. In this section, an algorithm to iteratively reach consensus is designed to assist the satellite dispatchers in adjusting their judgments so that they can reach the predefined consensus index. The detailed steps are shown in Algorithm 1.

Algorithm 1 Consensus Reaching Processes

Input: Individual satellite dispatcher e_k’s decision-making matrix $(R_{S}^{p})_{m \times n} = {r_{ij}^{p} | i = 1, \dots, m; j = 1, \dots, n; p = 1, \dots, f}$ with HF2TLSs, the parameter δ ∈ [0, 1], and the consensus threshold CON₀.

Output: Individual satellite dispatcher e_k’s decision-making matrix $(R_{S}^{p^{'}})_{m \times n}$ with HF2TLSs after reaching the consensus threshold, and and the ultimate degree of consensus CON (R).

Step 1. By Eq.(5), we can get the consensus degree CON (R) of initial decision-making matrix. If CON (R) > CON₀, output decision-making matrix with HF2TLSs; otherwise, go to the next step.

Step 2. Suppose ω_l (p, q) (p = 1, …, f, q = 1, …, f, p ≠ q) is the trust between any two satellite dispatchers, such that 0 ≤ ω_l (p, q) ≤1, ω_l (p, q) = ω_l (q, p) , ω_l (p, p) =1,total trust value of 1, The trust is derived from the consensus index. Calculating dispatcher trust ω_l (p, q) lays the foundation for consensus adjustment, as shown below: $ω_{l} (p, q) =$ (6) $\begin{matrix} \frac{1 - \frac{1}{m \times n} \sum_{i = 1}^{n} \sum_{j = 1}^{m} D (r_{ij}^{p}, r_{ij}^{q})}{\sum_{p = 1}^{f - 1} \sum_{q = p + 1}^{f} (1 - \frac{1}{m \times n} \sum_{i = 1}^{n} \sum_{j = 1}^{m} D (r_{ij}^{p}, r_{ij}^{q}))} \\ = & \frac{SIM (R_{S}^{p}, R_{S}^{q})}{\sum_{p = 1}^{f - 1} \sum_{q = p + 1}^{f} SIM (R_{S}^{p}, R_{S}^{q})}, \end{matrix}$

where $SIM (R_{S}^{p}, R_{S}^{q})$ is obtained from Eq.(4).

Step 3. CRPs: to locate which dispatcher needs to reevaluate his/her decision-making values. To find the decision-making matrix that should be adjusted, this paper follow two steps:

Identification of the position of the decision-making matrix(a):

Through Eq.(5), we know that the higher the consensus, the higher the similarity degree between satellite dispatchers. However, $SIM (R_{S}^{p}, R_{S}^{q})$ represents the similarity degree between any two dispatchers e_p and e_q, that is, there is going to be f relative values. In order to quickly identify the dispatcher who impacts the group consensus degree, let $SIM (R_{S}^{p}, R_{S}^{q}) = SIM (R_{S}^{q}, R_{S}^{p})$ , where p < q, p ≠ q. Therefore, the consensus degree of individual dispatcher is acquired as $\sum_{p = 1}^{f} SIM (R_{S}^{a}, R_{S}^{p})$ , and the position of the decision-making matrix is $POS = {(a) | S (R_{S}^{a}, R_{S}^{p}) = \min$ (7) ${\sum_{p = 1}^{f} SIM (R_{S}^{a}, R_{S}^{p}) | a = 1, 2, \dots, f, p \neq a}} .$

Adjustment of the decision-making matrix and iteration of consensus( $R_{S}^{a}$ ):

Based on the dispatcher¡^– trust and the decision-making matrices of other members, the decision-making information adjustment function is designed to achieve joint management. $(R_{S}^{p^{'}})_{m \times n} = δ R_{S}^{a} + (1 - δ) (\sum_{q = 1, q \neq a}^{f} ω_{l} (a, q) E (R_{S}^{q})) .$ (8)

Step 4. A novel decision-making matrix recalculates the consensus index is by Eq.(5), return to step 1 for judgment, iteration, and adjustments until the consensus threshold is reached.

Step 5. End.

4 The framework of satellite emergency mission scheduling scheme group decision-making support model

In this section, we give the detailed steps of the method framework to address the SEMSSGDM problem. The framework consists of three algorithms as shown in Fig. 1. Algorithm 1 is described in detail in Section 3.3. Algorithm 2 and Algorithm 3 will be described in this section.

Fig.1

The framework of satellite emergency mission scheduling scheme group decision-making support model.

Let X = {x₁, x₂, …, x_m} be a set of m alternatives, C = {c₁, c₂, …, c_n} be a collection of n alternatives satellite scheme indicators, and E = {e₁, e₂, …, e_f} be an assembly of satellite dispatchers. Each satellite dispatcher e_p offers his/her individual HF2TLS $(R_{S}^{p})_{m \times n} = {r_{ij}^{p} | i = 1, \dots, m; j = 1, \dots, n}$ , $r_{ij}^{p} = {(s_{ι}^{p}, α_{ι}^{p}) | ι = 1, 2, \dots, l (r_{ij}^{p})}$ on S = {s₀, s₁, …, s_g} based on several chosen indicators. As shown in Fig. 1, this method framework involves the following steps:

Obtain individual satellite dispatcher e_k’s SEMSSGDM information evaluation matrix according to schemes results, and transform the linguistic expression into HF2TLSs: $(R_{S}^{p})_{m \times n} = {r_{ij}^{p} | i = 1, \dots, m; j = 1, \dots, n; p = 1, \dots, f}$ .

CRPs: apply Algorithm 1 consensus to calculate the values of $(R_{S}^{p^{'}})_{m \times n}$ and CON (R), and make different dispatchers to come to a consensus.

Aggregation process: aggregate individual SEMSSGDM matrix $(R_{S}^{p^{'}})_{m \times n}$ into X_m×n using Algorithm 2 to obtain a consensus group matrix. The aggregation process is the beginning of the scheme selection process.

Scheme ranking processes: mapping the three subordinate ranking methods of RS, RP and FMF by Algorithm3 to obtain the final ranking of the alternatives and improve the accuracy and robustness of the scheme ranking results. The scheme ranking processes is the finishing step of the scheme selection processes.

Algorithm 2 Aggregation Processes–PSO

Input: The SEMSSGDM matrix $(R_{S}^{p^{'}})_{m \times n}$ with HF2TLSs after reaching the consensus threshold of individual satellite dispatcher e_k.

Output: The final matrix X_m×n iterated and aggregated by the PSO algorithm.

Step 1. Find the maximum and minimum values of all HF2TLSs decision-making matrices e_k at each position, thus forming an interval matrix formed by two matrices P₁ and P₂. $P_{1} = \min {r_{ij}^{(1)}, r_{ij}^{(2)}, \dots, r_{ij}^{(m + n)}},$ (9) $P_{2} = \max {r_{ij}^{(1)}, r_{ij}^{(2)}, \dots, r_{ij}^{(m + n)}} .$ (10)

Step 2. Initialize particle swarm: given population size N, randomly generate the position X_i and velocity V_i of each particle. One can see that all the matrices X_i belong to ([P₁, P₂]) _m×n.

Step 3. The fitness value of each particle is calculated by the objective function f (x): $For i = 1 to N$ (11) $\begin{matrix} fitness [i] = D_{e} (R_{S}^{p}, R_{S}^{q}) \\ = & \frac{1}{m \times n} \sum_{k = 1}^{m} \sum_{j = 1}^{n} D (r_{kj}^{1}, r_{kj}^{2}) \\ = & \frac{1}{m \times n} \sum_{k = 1}^{m} \sum_{j = 1}^{n} [max {max_{s_{ι}^{p} \in r_{kj}^{1}} min_{s_{ι}^{q} \in r_{kj}^{2}} | \frac{(s_{ι}^{p} + α_{ι}^{p} - (s_{ι}^{q} + α_{ι}^{q})}{g} |, \\ max_{s_{ι}^{q} \in r_{kj}^{2}} min_{s_{ι}^{p} \in r_{kj}^{1}} | \frac{(s_{ι}^{q} + α_{ι}^{q}) - (s_{ι}^{p} + α_{ι}^{p}}{g} |}] \\ = & f (X [i]) \\ Next i . \end{matrix}$

Step 4. The best position (local best) is obtained for each particle under the objective function f (x): $\begin{matrix} For i = 1 to N \\ pBest [i] \leftarrow f (x); \\ Next i . \end{matrix}$

Step 5. Obtain the position of the best particle (global best) of the entire population (the best particle value is between P₁ and P₂) by the objective function f (x): $\begin{matrix} gBest \leftarrow f (x) . \end{matrix}$

Step 6. Update the velocity V_i and position X_i for each particle. For convenience, refer to literature [49] for the specific function.

Step 7. If the termination conditions are met, exit the algorithm; otherwise, return to Step 3.

Algorithm 3 Scheme Ranking Processes–MULTIMOORA with HF2TLSs

Input: The final decision-making matrix R_m×n iterated and aggregated by PSO algorithm.

Output: The sequence results of satellite alternatives under RS, RP, and FMF.

Step 1. Standardize final decision-making matrix X = (x_ij) _m×n, and obtain the dimensionless value $\tilde{X} = ({\tilde{x}}_{ij})_{m \times n}$ . ${\tilde{x}}_{ij} = \frac{▵^{- 1} (x_{ij})}{\sqrt{\sum_{i = 1}^{m} (▵^{- 1} (x_{ij}))^{2}}} .$ (12)

Step 2. Compute the utility values ${\tilde{u}}_{RS} (i = 1, 2, \dots, m)$ by the RS model based on the below arithmetic integration. ${\tilde{u}}_{RS} = max_{i} {\sum_{j = 1}^{g} {\tilde{x}}_{ij} - \sum_{j = g + 1}^{n} {\tilde{x}}_{ij}},$ (13)

where g and n - g, respectively, represent the number of benefit-type and cost-type attributes[39]. Derive the first subordinate ranking in descending order of ${\tilde{u}}_{RS}$ [41].

Step 3. Calculate the utility values ${\tilde{u}}_{RP} (i = 1, 2, \dots, m)$ by the RP model. The main idea of this model is to find the maximum distance between the maximum value of each attribute and each attribute of the alternative ${\tilde{u}}_{RP} = max_{i} ∣ r_{j} - {\tilde{x}}_{ij} ∣,$ (14)

where $r_{j} = max_{i} {\tilde{x}}_{ij}$ . The alternatives are ranked in ascending order of ${\tilde{u}}_{RP}$ [41].

Step 4. Determine the utility values ${\tilde{u}}_{FMF} (i = 1, 2, \dots, m)$ by the FMF model on the basis of geometric integration function (15). Then, obtain the third subordinate ranking in descending order of ${\tilde{u}}_{FMF}$ [41]. ${\tilde{u}}_{FMF} = max_{i} {\frac{\prod_{j = 1}^{g} {\tilde{x}}_{ij}}{\prod_{j = g + 1}^{n} {\tilde{x}}_{ij}}} .$ (15)

Step 5. Obtain the final ranking of alternatives by the dominance theory [50].

5 An application example based on SEMSSGDM

To verify the feasibility of the proposed decision-making method, an application example is applied to illustrate it. Firstly, multiple sets of emergency mission scheduling schemes are generated according to the mission scheduling intelligent decision system, schemes of which is shown in Fig. 2, and then the SEMSSGDM method is adopted for scheme decision making. According to the observation requirements of emergency missions, the mission scheduling system generates four sets of emergency scheduling schemes x_i (i = 1, 2, 3, 4) as alternative schemes, among which x₁ stresses on the emergency time window, whereby the more observation time windows, the more missions are observed earlier; x₂ concerns the completion number, whereby more emergency missions are scheduled if possible; x₃ emphasizes the emergency timeliness, whereby early missions are observed immediately; and x₄ focuses on disturbance, whereby minor disturbances to the original mission are observed initially. In different emergency missions, dispatchers have different identities, and for the evaluation of emergency schemes, different levels of dispatchers are needed to carry out the evaluation. Therefore, three levels of dispatchers were selected and the panel consisted of three dispatchers, which are experienced chiefse₁, frontline intelligence officers e₂ with quick access to information, and duty leaders e₃ with strong management skills.

Fig.2

Results of emergency mission scheduling schemes in mission scheduling intelligent decision making simulation system.

With the time-limited, heterogeneous and dynamic nature of the satellite emergency missions, the emergency mission time window is urgently needed and the mission can be completed quickly under the limited time window, thus requiring considering the mission completion when selecting indicators; Moreover, multiple types of resources such as satellites, ground stations and extensive observation missions in the original scheme are involved in the scheduling, and require the rapid arrangement of emergency missions into the original scheme and minimize the disturbance to the original scheme under the handling of resource constraints, time window constraints, etc. Accordingly, the SEMSSGDM indicators are segmented into three effectiveness indicators, which are emergency mission completion, scheme performance and resource utilization capability [51].

5.1 An application example of the proposed methods

Consequently, we employed questionnaires to derive dispatcher decision information for four schemes across three indicators supported by the scheduling results of the mission scheduling intelligent decision system. Three satellite dispatchers e_i(i = 1, 2, 3) are asked to conduct pairwise comparisons of the four schemes in a hesitant fuzzy 2-tuple linguistic environment. Typically, linguistic label sets are pre-defined for reference when evaluating all alternatives. In this context, seven linguistic labels are adopted: S = {s₀ = extremely bad, s₁ = very bad, s₂ = bad, s₃ = general, s₄ = good, s₅ = very good, s₆ = extremely good}.

Obtain individual SEMSSGDM opinions HF2TLSs $(R_{S}^{p}) = (r_{ij}^{p})_{4 \times 3} (p = 1, 2, 3)$ of different dispatcher through questionnaires. These matrices are shown in Tables 1,2 , and 3, respectively.

Apply Eq.(2) to calculate $D^{k} (R_{S}^{p}, R_{S}^{q})_{4 \times 3}$ , where k = 1, 2, 3 respectively represent Hausdorff distances between decision-making matrices $R_{S}^{1}$ and $R_{S}^{2}$ , $R_{S}^{2}$ and $R_{S}^{3}$ , and $R_{S}^{1}$ and $R_{S}^{3}$ . The distances are given below: $\begin{matrix} D^{1} = (\begin{matrix} 0.4500 & 0.3500 & 0.0833 \\ 0.1167 & 0.2667 & 0.2333 \\ 0.0500 & 0.0417 & 0.3167 \\ 0.3500 & 0.1833 & 0.0167 \end{matrix}), \\ D^{2} = (\begin{matrix} 0.6500 & 0.4083 & 0.2500 \\ 0.3750 & 0.0667 & 0.3333 \\ 0.5000 & 0.0167 & 0.1667 \\ 0.3000 & 0.3333 & 0.1167 \end{matrix}), \\ D^{3} = (\begin{matrix} 0.3167 & 0.2083 & 0.3333 \\ 0.2583 & 0.2667 & 0 \\ 0.4500 & 0.0583 & 0.0250 \\ 0.0333 & 0.0417 & 0.1000 \end{matrix}) . \end{matrix}$

Obtain similarity degree $SIM (R_{S}^{p}, R_{S}^{q})$ between decision-making matrices $R_{S}^{1}$ and $R_{S}^{2}$ , $R_{S}^{2}$ and $R_{S}^{3}$ , and $R_{S}^{1}$ and $R_{S}^{3}$ by Eq.(4). Consequently, resulting in $SIM (R_{S}^{1}, R_{S}^{2} = 0.7951, SIM (R_{S}^{2}, R_{S}^{3} = 0.7069, SIM (R_{S}^{1}, R_{S}^{3} = 0.8257$ .

Calculate the consensus degree by Eq.(5). Then, CON (R) =0.7759, and CON₀ = 0.8. We can find that the consensus is unacceptable. Therefore, we must adjust the decision-making matrices. First, we identify the position of the dispatcher’s decision-making matrix that should be adjusted by Eq.(7); thus, we can obtain S = 1.5021, and POS = 2. Second, we utilize Eq.(6) to calculate ω_l (1, 2) =0.3401 ; ω_l (2, 3) =0.3140. Finally, we suppose δ = 0.7, and thus the decision-making matrix adjusted with Eq.(8) is shown in Table 4.

In the same way, Algorithm 1 consensus is iterated until the consensus reaches the threshold CON₀ = 0.8. When CON₀ = 0.8013, the number of adjustments k = 5, and the three final decision-making matrices are as shown in Tables 5,6 , and 7, respectively.

According to 9 and 10, an interval-valued comparison matrix is constituted by P₁ and P₂.

$\begin{matrix} P_{1} = (\begin{matrix} (s_{3}, - 0.25) & (s_{1}, 0.4954) & (s_{2}, 0.4) \\ (s_{1}, 0.4220) & (s_{4}, - 0.1471) & (s_{2}, - 0.2288) \\ (s_{0}, 0.4) & (s_{4}, 0.2184) & (s_{3}, 0) \\ (s_{4}, - 0.3922) & (s_{3}, 0.1) & (s_{2}, 0.4) \end{matrix}), \\ P_{2} = (\begin{matrix} (s_{6}, 0) & (s_{3}, 0.45) & (s_{4}, 0.4) \\ (s_{3}, 0.4) & (s_{6}, 0) & (s_{4}, - 0.2) \\ (s_{4}, - 0.2) & (s_{5}, 0.35) & (s_{4}, 0) \\ (s_{6}, - 0.1) & (s_{4}, 0.0536) & (s_{3}, 0) \end{matrix}) . \end{matrix}$

Apply Algorithm 2 to aggregate individual decision-making values. We set N = 500 iterations. The final opinions of all of the satellite dispatchers between P₁ and P₂ are given in Table 8. Moreover, the fitness curve based on PSO fitness function is shown in Fig. 3. The fitness function values are based on 500 iterations in PSO, and we can see that the fitness function is slowly minimized, with the optimum value at 5.26e-07.

Standardize the matrix X_m×n to $\tilde{X}$ , three sorting results of methods RS, RP and FMF are then mapped through Algorithm 3. The final ranking result is obtained based on the dominance theory, as shown in Table 9. This thus improves the accuracy and robustness of the results. $\begin{matrix} \tilde{X} = (\begin{matrix} 2.5000 & 1.4375 & 1.0000 \\ 1.4167 & 1.9583 & 1.3750 \\ 1.2917 & 2.0833 & 1.2500 \\ 2.4583 & 1.5625 & 1.0000 \end{matrix}) . \end{matrix}$

Rank schemes. From the previous step, we have x₄ > x₂ > x₁ > x₃. Thus, the optional satellite emergency mission scheduling scheme is x₄.

Table 1
Individual SEMSSGDM matrix $R_{S}^{1}$ with HF2TLSs

c ₁ c ₂ c ₃

x ₁ { (s₄, 0.1) , (s₅, - 0.2)} { (s₂, 0.2) , (s₃, 0.1)} { (s₄, 0.4)}

x ₂ { (s₂, - 0.15)} { (s₅, 0.25) , (s₆, 0.3)} { (s₃, 0.3)}

x ₃ { (s₀, 0.4)} { (s₅, 0.35)} { (s₃, 0.1) , (s₄, - 0.15)}

x ₄ { (s₅, 0.25) , (s₆, - 0.3)} { (s₄, 0)} { (s₃, 0)}

	c ₁	c ₂	c ₃
x ₁	{ (s₄, 0.1) , (s₅, - 0.2)}	{ (s₂, 0.2) , (s₃, 0.1)}	{ (s₄, 0.4)}
x ₂	{ (s₂, - 0.15)}	{ (s₅, 0.25) , (s₆, 0.3)}	{ (s₃, 0.3)}
x ₃	{ (s₀, 0.4)}	{ (s₅, 0.35)}	{ (s₃, 0.1) , (s₄, - 0.15)}
x ₄	{ (s₅, 0.25) , (s₆, - 0.3)}	{ (s₄, 0)}	{ (s₃, 0)}

Table 2

Individual SEMSSGDM matrix $R_{S}^{2}$ with HF2TLSs

	c ₁	c ₂	c ₃
x ₁	{ (s₂, 0.1)}	{ (s₁, 0)}	{ (s₃, 0.4) , (s₄, - 0.1)}
x ₂	{ (s₁, 0.15)}	{ (s₄, 0.3) , (s₅, - 0.3)}	{ (s₁, 0.3) , (s₂, - 0.1)}
x ₃	{ (s₀, 0.1) , (s₁, - 0.2)}	{ (s₅, 0.1)}	{ (s₅, 0)}
x ₄	{ (s₄, - 0.4)}	{ (s₅, 0.1) , (s₆, - 0.25)}	{ (s₃, 0.1)}

Table 3

Individual SEMSSGDM matrix $R_{S}^{3}$ with HF2TLSs

	c ₁	c ₂	c ₃
x ₁	{ (s₆, 0)}	{ (s₃, 0.45)}	{ (s₂, 0.4)}
x ₂	{ (s₃, 0.4)}	{ (s₅, - 0.3)}	{ (s₃, 0.3) , (s₄, - 0.2)}
x ₃	{ (s₃, 0.1) , (s₄, - 0.2)}	{ (s₅, 0)}	{ (s₃, 0) , (s₄, 0)}
x ₄	{ (s₅, 0.4) , (s₆, - 0.1)}	{ (s₃, 0.1) , (s₄, - 0.25)}	{ (s₂, 0.4)}

Table 4

The adjusted SEMSSGDM matrix $R_{S}^{2^{'}}$

	c ₁	c ₂	c ₃
x ₁	{ (s₃, - 0.0503)}	{ (s₂, - 0.3548)}	{(s₃, - 0.3579), (s₃, - 0.1864)}
x ₂	{ (s₂, 0 . -0.4920)}	{(s₄, - 0.2684), (s₄, - 0.1312)}	{(s₂, - 0.0863), (s₂, 0.1195)}
x ₃	{(s₁, - 0.1651), (s₁, 0.0750)}	{ (s₄, - 0.0269)}	{ (s₃, 0.2115)}
x ₄	{ (s₄, - 0.3795)}	{(s₃, 0.3474), (s₄, - 0.4297)}	{ (s₂, 0.2271)}

Table 5

Individual SEMSSGDM matrix $R_{S}^{1^{*}}$ with HF2TLSs

	c ₁	c ₂	c ₃
x ₁	{ (s₄, 0.1) , (s₅, - 0.2)}	{(s₂, 0.2), (s₃, 0.1)}	{ (s₄, 0.4)}
x ₂	{ (s₂, - 0.15)}	{(s₅, 0.25), (s₆, 0.3)}	{ (s₃, 0.3)}
x ₃	{ (s₀, 0.4)}	{ (s₅, 0.35)}	{(s₃, 0.1), (s₄, - 0.15)}
x ₄	{(s₅, 0.25), (s₆, - 0.3)}	{ (s₄, 0)}	{ (s₃, 0)}

Table 6

Individual SEMSSGDM matrix $R_{S}^{2^{*}}$ with HF2TLSs

	c ₁	c ₂	c ₃
x ₁	{ (s₃, - 0.2500)}	{ (s₁, 0.4954)}	{(s₃, - 0.1904), (s₃, 0.0546)}
x ₂	{ (s₁, 0.4220)}	{(s₄, - 0.1471), (s₄, 0.0489)}	{(s₂, - 0.2288), (s₂, 0.0652)}
x ₃	{(s₁, - 0.3365), (s₁, 0.0065)}	{ (s₄, 0.2184)}	{ (s₄, - 0.3933)}
x ₄	{ (s₄, - 0.3922)}	{(s₄, - 0.2649), (s₄, 0.0536)}	{ (s₃, 0.4194)}

Table 7

Individual SEMSSGDM matrix $R_{S}^{3^{*}}$ with HF2TLSs

	c ₁	c ₂	c ₃
x ₁	{ (s₆, 0)}	{ (s₃, 0.45)}	{ (s₂, 0.4)}
x ₂	{ (s₃, 0.4)}	{ (s₅, - 0.3)}	{ (s₃, 0.3) , (s₄, - 0.2)}
x ₃	{ (s₃, 0.1) , (s₄, - 0.2)}	{ (s₅, 0)}	{ (s₃, 0) , (s₄, 0)}
x ₄	{ (s₅, 0.4) , (s₆, - 0.1)}	{ (s₃, 0.1) , (s₄, - 0.25)}	{ (s₂, 0.4)}

Table 8

The collective opinion for all satellite dispatchers

	c ₁	c ₂	c ₃
x ₁	{ (s₆, 0)}	{ (s₃, 0.45)}	{ (s₂, 0.4)}
x ₂	{ (s₃, 0.4)}	{ (s₅, - 0.3)}	{ (s₃, 0.3)}
x ₃	{ (s₃, 0.1)}	{ (s₅, 0)}	{ (s₃, 0)}
x ₄	{ (s₆, - 0.1)}	{ (s₄, - 0.25)}	{ (s₂, 0.4)}

Fig.3

Fitness curve.

Table 9

Results derived by the MULTIMOORA method

	RS		RP		FMF		Final
	${\tilde{u}}_{RS}$	Rank	${\tilde{u}}_{RP}$	Rank	${\tilde{u}}_{FMF}$	Rank	rank
x ₁	4.9375	2	1.4583	3	3.5938	3	3
x ₂	4.7500	3	1.0833	1	3.8147	2	2
x ₃	4.6250	4	1.2083	2	3.3637	4	4
x ₄	5.0208	1	1.4583	3	3.8412	1	1

The matrix adjustment parameter is used to update the decision-making matrix that influences group consensus in Eq.(8). The value of the matrix adjustment parameter δ ranges from 0 to 1 with a step of 0.1. The number of iterations is 500. The computational statistics of the simulation are shown in Fig. 4. From the figure, it can be seen that, all other conditions being equal, the value of the consensus degree CON(R) is better when the parameter δ is between the interval [0.5, 0.9], and reaches its maximum when the parameter δ = 0.7. And when δ < 0.5 the consensus degree is less effective. Evidently, according to the decision-making information adjustment function Eq.(8) in Algorithm 1 consensus, it is better to work with consensus when making even small changes to one’s own decision matrix based on the original matrix based on other dispatchers’ opinion matrices, rather than just dismissing non-consensus opinions.

Fig.4

Consensus value CON(R) of different parameter δ.

5.2 Comparison analysis and discussion

To validate the effectiveness of the designed method, it is compared with other methods, primarily with the literature [52 –54] for three comparative analyses. The method proposed by Wang et al. [52] (The scheme parameter in [52] is represented by r) is based on the hesitant 2-tuple linguistic correlated averaging (H2TLCA) aggregation operator and the hesitant 2-tuple linguistic correlated geometric (H2TLCG) aggregation operator. The order of the solutions is r₄ > r₁ > r₂ > r₃. Meanwhile, we can obtain a final ranking of r₄ > r₁ = r₂ > r₃ by applying our method. The comparison is shown in Table 10. The results are different because their method ignores the subjective influence of experts on the aggregation process. Simultaneously, the group consensus level is not considered, which results in lower satisfaction with the obtained result. We also made a comparison to the method presented by Wang et al. [53] (The scheme parameter in [53] is represented by t), which used the generalized 2-tuple linguistic weighted averaging (G2TLWA) operator to aggregate the decision-making matrices under multi-hesitant fuzzy linguistic term sets. The result obtained by this method is t₅ > t₄ > t₂ > t₁ > t₃, while that obtained by our method is t₅ > t₄ > t₁ > t₂ > t₃. These results are presented in Table 11. There is a little difference in the order of t₁ and t₂ because the dispatchers’ relative weights are custom rather than calculated in the compared method, and therefore subjectivity affects the results. Moreover, the compared method does not consider group consensus.

Table 10
Results derived by our method in [52]

RS RP FMF Final

${\tilde{u}}_{RS}$ Rank ${\tilde{u}}_{RP}$ Rank ${\tilde{u}}_{FMF}$ Rank rank

x ₁ 4.0000 2 0.6667 1 2.0000 2 2

x ₂ 4.0000 2 0.6667 1 2.0000 2 2

x ₃ 3.6667 3 0.6667 1 1.6667 3 3

x ₄ 4.3333 1 0.6667 1 2.7778 2 1

	RS	RP	FMF	Final
x ₁	4.0000	2	0.6667	1	2.0000	2	2
x ₂	4.0000	2	0.6667	1	2.0000	2	2
x ₃	3.6667	3	0.6667	1	1.6667	3	3
x ₄	4.3333	1	0.6667	1	2.7778	2	1

Table 11

Results derived by our method in [53]

	RS		RP		FMF		Final
	${\tilde{u}}_{RS}$	Rank	${\tilde{u}}_{RP}$	Rank	${\tilde{u}}_{FMF}$	Rank	rank
t ₁	11.4774	1	3.4917	4	23.9422	3	3
t ₂	9.4902	4	3.4928	5	12.4781	4	4
t ₃	8.9756	5	2.9959	3	11.8889	5	5
t ₄	9.9891	3	2.9942	2	24.3678	2	2
t ₅	10.4755	2	2.9938	1	27.8068	1	1

The Hausdorff distance under HF2TLSs presented in this paper can directly calculate the distance between two HF2TLS without considering whether the length is equal, thereby eliminating the disadvantages caused by increasing or decreasing the length of point sets for decision-makers. To confirm this advantage, we applied this distance to the method proposed by [54], which also considered the CRPs. The ranking is x₃ > x₂ > x₁ > x₄. The results are shown in Table 12. This is the same as the optimal ranking after comparing [54] with other methods, which illustrates the effectiveness of the method used in this paper. Xu and Wang [54] established a decision support model through personal consistency, and constructed a consensus model based on the Manhattan distance. The Manhattan distance is mostly used to calculate the distance between points; it cannot be applied to calculate the distance between collections, and cannot consider different lengths. Although the method presented by Xu and Wang is based on hesitant fuzzy 2-tuple linguistic decision-making information, it fails to directly calculate the group consensus of an uncertain set length model. Hence, results are not accurate. It can be said that the consensus model adopted in the present paper considers more comprehensive factors and provides more accurate and effective decision-making results based on different decision-making information. On the other hand, the SEMSSGDM information is aggregated based on PSO and MULTIMOORA to ensure the accuracy and robustness of the decision-making results. Moreover, we examined some literature based on Hausdorff distance in the context of hesitant fuzzy 2-tuple linguistic sets, such as the study by Wang and Wu et al.[28], which used the Hausdorff distance in other aggregation methods but still did not consider consensus. There is limited published work on establishing a consensus model based on the Hausdorff distance in hesitant fuzzy 2-tuple linguistic set. In contrast, there is little literature on the SEMSSGDM problem using HF2TLS to study it and to build a consensus model, so this aspect of the comparative analysis will not be undertaken.

Table 12

Results derived by our method in [54]

	RS		RP		FMF		Final
	${\tilde{u}}_{RS}$	Rank	${\tilde{u}}_{RP}$	Rank	${\tilde{u}}_{FMF}$	Rank	rank
x ₁	3.5003	3	1.5003	2	0.2813	3	3
x ₂	4.2507	2	1.5003	2	0.5472	2	2
x ₃	5.0004	1	1.0002	1	1.9694	1	1
x ₄	3.5003	3	1.5003	2	0.2188	4	4

According to the above comparative analysis, the method proposed in this paper has the following advantages for the SEMSSGDM problem of HF2TLSs.

First, the HF2TLSs adopted in this paper can express the evaluation information with more flexibility. They can well characterize the linguistic information that is hesitantly fuzzy for multiple emergency schemes decision-making, preserving the integrity of the original data or the intrinsic thoughts of the dispatcher, which is a prerequisite to ensure the accuracy of the final results.

Secondly, the proposed Hausdorff distance of HF2TLSs differs from existing distance measures by adding values to sets with fewer elements, whereas the presented distance measure automatically computes the distance between HF2TLSs directly without additional operations.

Lastly, the proposed Hausdorff distance of HF2TLSs differs from existing distance measures by adding values to sets with fewer elements, whereas the presented distance measure automatically computes the distance between HF2TLSs directly without additional operations. Furthermore, the proposed method framework in this paper utilizes different methods in different decision stages, thus ensuring the accuracy and robustness of the results.

6 Conclusions

Group consensus is an important research subject for the practical SEMSSGDM. Based on the analysis of the characteristics of satellite emergency decision environment, an improved method for SEMSSGDM problems is presented, which focuses on constructing a new consensus model in HF2TLS-based. The major contributions of this paper are as follows:

Developed a non-normalised Hausdorff distance function to calculate the distance between two HF2TLS. The HF2TLS expansion-based Hausdorff distance can ascertain the distance between any two HF2TLSs with different number of elements, which is not influenced by the subjective preference of the dispatcher.

Constructed a novel consensus model with Hausdorff distance. The model accounts for the degree of trust between satellite dispatchers. In each CPR, the first step is to identify the dispatcher with the most divergent opinion from the group. The identified dispatcher is then asked to adjust his or her individual opinion. The adjustment is directed by the other dispatchers.

Combined PSO and MULTIMOORA methods for increasing the accuracy of the SEMSSGDM selection process results and optimising the scheme selection process. First, the PSO algorithm used the novel fitness function to aggregate the adjusted individual SEMSSGDM matrix. We proposed the MULTMOORA method under HF2TLSs, which performs three sorts of the optimal global matrices found by PSO, and obtains three sorting results by three subordinate methods: RS, RP, and FMF. This method then used the dominance theory to produce the final sorting SEMSSGDM result.

Finally, we conducted three comparative analyses to illustrate the advantages of the developed method. The results showed that our evaluation method is more flexible, practical and useful for evaluating and sequencing satellite emergency mission schemes in a context of uncertainty. The suggested methodology may also be applied to other fields. Furthermore, the proposed decision-making method still has some shortcomings and limitation, pointing out future research directions.

First of all, we consider the consensus problem of small group decision making, but in actual group decision making, large-scale group decision making(LSGDM) is more worthy of attention, especially in emergency decision making. Therefore, in future work, we will carry out decision-making of satellite emergency mission scheduling scheme based on LSGDM.

Third, our SEMSSGDM indicators system is relatively simple. There are many factors affecting the efficiency of satellite emergency mission scheduling. A multi-scale and multi-dimensional indicator system should be established to provide more reasonable decision results.

In follow-up work, we will address the shortcomings and limitation mentioned above, evaluate the scheme closer to actual engineering problems, and provide feedback and support for satellite mission scheduling.

Footnotes

Acknowledgments

This paper is supported by the Natural Science Foundation of China (nos. 72071064, 71521001, 71871079).

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An improved method for satellite emergency mission scheduling scheme group decision-making incorporating PSO and MULTIMOORA

Abstract

Keywords

Abbreviation

1 Introduction

2 Preliminaries

2.1 Hesitant fuzzy 2-tuple linguistic set

3 Novel consensus reaching model considering the distance and consensus degree

3.1 The Hausdorff distance for HF2TLSs

Table 10 Results derived by our method in [52] RS RP FMF Final u ˜ RS Rank u ˜ RP Rank u ˜ FMF Rank rank x 1 4.0000 2 0.6667 1 2.0000 2 2 x 2 4.0000 2 0.6667 1 2.0000 2 2 x 3 3.6667 3 0.6667 1 1.6667 3 3 x 4 4.3333 1 0.6667 1 2.7778 2 1

Footnotes

Acknowledgments

References

Table 10
Results derived by our method in [52]

RS RP FMF Final

${\tilde{u}}_{RS}$ Rank ${\tilde{u}}_{RP}$ Rank ${\tilde{u}}_{FMF}$ Rank rank

x ₁ 4.0000 2 0.6667 1 2.0000 2 2

x ₂ 4.0000 2 0.6667 1 2.0000 2 2

x ₃ 3.6667 3 0.6667 1 1.6667 3 3

x ₄ 4.3333 1 0.6667 1 2.7778 2 1