# Spherical Fuzzy Decision-Making Approach Integrating Delphi and TOPSIS for Package Tour Provider Selection

Authors: Phi-Hung Nguyen
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The spherical fuzzy sets were recently developed among various fuzzy sets to handle the hesitancy representation issue in multiple criteria decision-making (MCDM) problems, where experts provide information about attributes in the form of spherical fuzzy numbers using linguistic variables. The main purpose of this study is to develop a novel approach integrating Delphi Technique for Order of Preference by Similarity to Ideal Solution based on spherical fuzzy sets (SF-Delphi and SF-TOPSIS). First, the SF-Delphi technique is suggested to derive a valid set of critical criteria based on qualitative information and linguistic preferences. Second, the SF-TOPSIS approach is utilized to rank alternatives based on different spherical fuzzy aggregating operators. Hence, to validate the effectiveness of the proposed methodology, an empirical case study of package tour provider selection is given. Seventeen critical criteria related to four main dimensions (price, service quality, information and technology, and location) were shortlisted and validated from literature and expert opinions. Ten potential package tour providers from Vietnam were ranked in this study. A comparative analysis was conducted to check the proposed methodology’s robustness and validity. The results indicated that the novel SF-Delphi technique may become very helpful for dealing with critical factors, and SF-TOPSIS could be applied to decision problems in uncertain data environments. Furthermore, this research’s findings imply that tour operators should emphasize the most critical attributes to increase the appeal and competitiveness of their package tour products.


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## Body

## 1. Introduction

Package tours have been introduced to tourists as an economical and less preparation effort option for a long time. Package tours vary by destination and by each travel agency; customers can have different choices from luxury to budget price, which may include flight tickets, meals, accommodation, and visiting tickets [1–3]. Mainly focusing on clients’ experience and service at destinations, tourism is typically an industry that can hardly be replaced. The previous study proved that travelers’ decision is affected by many aspects. Several methods were applied to identify the determinants factors, helping travel agencies and authorities to adjust their marketing strategy and policies to attract more tourists [4]. However, after the COVID-19 pandemic, travel needs are predicted to increase considerably after many countries’ extended quarantine and border-closed [5]. Distancing society and lockdown led to the decrease of the economy, not only the finance but also the physical and mental health of many people [6]. Even being urged by traveling desire, choice, and customer decision may be driven by factors that are not the same as before the pandemic. Affected by the COVID-19 [7], international arrival to Vietnam dropped by nearly 15 million to only 3.7 million arrivals up to March 2020, when Vietnam closed the borders to tourists. Slightly better than international tourism, national tourist numbers also decreased from 62 million in 2019 to 46 million in 2020. The hotels’ occupancy rate felt significant to only 30% on average [8].Although not seriously affected by the epidemic as many other countries, Vietnam did have a long-time border closing to neighboring countries. The short-term social lockdown in 2020 also led to the downturn of much domestic business; the number of bankruptcies also increased significantly. Vietnam National Administration of Tourists reported that international arrivals to Vietnam in 2020 reduced to 78.7% and focused more on domestic tourism. It is necessary to define the right attractive factors for tourists to create suitable products and services to compete in the market. Moreover, after the Covid-19, it will take time for the economy to rehabilitate. Hence, having an all-in-one package will be an excellent solution for the tourists’ budget in this financially sensitive period.Package tours combine various services; multiple studies have been conducted to determine how the various components contribute to a better package [9]. Regarding the study of [10], the provided schedule, price, friends’ suggestions, and the departure dates’ suitability affected the tourists’ choice of the package tour. Product-related factors and travelers' behaviors also significantly influence the selection [11]. According to [12], persons who travel alone have different needs and interests than those who travel with relatives or friends. Package tour travelers may have different, accompanied people like their spouses, family members, or friends. Family group travelers often consider facilities, safety, and accessibility essential factors different from individual travelers [13]. First-time travelers contribute largely to package tour customers while seasoned travelers are few [14]. Fesenmaier et al. [15] divided visitor preferences into personal and travel characteristics. Personal traits include socioeconomic status, education, and employment and demographic information, such as age and gender. These factors are predicted to pertain to tourist preferences. Kanellopoulos [16] indicated that when planning a group package tour, tourists consider many factors, including the type of group package tour, destination country, departure date, accommodation and food quality, optional activities, tour cost and length, sport and nonsport activities, travel risk, and transportation.Numerous studies have been conducted on consumer behavior and decision-making in various fields, including manufacturing, finance, logistics, and supply chain management [17–22]. By comprehending the guests’ desires and expectations, organizations may develop an efficient marketing plan, offer the appropriate items, and improve the quality of their service to suit the tourists’ requirements. In general, past studies employed a variety of research approaches. However, selecting a package tour is a complicated decision influenced by various factors relating to the services and travelers. To gain a thorough grasp of this topic, a multidimensional framework is required. When the study issue is complicated and uncertain among the multiple approaches available. MCDM is regarded as a practical approach [23–26]. The domain of MCDM has advanced dramatically in recent years, owing to many academic and scientific publications devoted to applying specialized decision-making models in a given field. Additionally, MCDM is a convenient tool for resolving complicated problems since it evaluates various options using a specific set of criteria.The primary motivation for scholars to develop MCDM is to establish mathematical formulas that will aid in evaluating criteria and selecting the most appropriate alternative [27]. Fuzzy set theory is accelerating its spread across all science disciplines [28–31]. More than any other branch, control theory, and decision-making extensively use the fuzzy set theory. Each new extension of fuzzy sets creates a new opportunity for researchers to use them to further their research topics. Intuitionistic, hesitant, picture, and Pythagorean fuzzy sets have developed into a considerable extension, from which practically all subsequent extensions are derived [32, 33]. Spherical fuzzy sets are predicted to be preferred by most academics in the near future, as their principles are sufficiently solid to support further development [34–36]. Spherical fuzzy sets (SFS) were proposed by [37, 38], which extended intuitionistic, Pythagorean, neutrosophic, and picture fuzzy sets. SFS enables decision-makers to generalize various extensions of fuzzy sets by creating a membership function on a spherical surface and independently allocating its parameters to a broader domain.The evaluation facts given by experts are frequently fuzzy and imprecise due to the inherent uncertainty and vagueness, ambiguity, and subjectivity of information in a complex decision-making environment. While MCDM models have been developed to aid decision-makers in picking travel packages, few examine the issue in a fuzzy decision-making environment and developing countries. According to a recent assessment, tourism contributes around 9.2% to Vietnam's GDP [39]; although the term package tour has been widely used in the tourism industry, there is a shortage of literature on consumer behavior using MCDM approaches in the Vietnam market for package tour. In growing markets such as Vietnam, no study has been undertaken to establish the essential aspects impacting consumers’ package tour purchasing behavior and rank package tour providers using MCDM techniques based on spherical fuzzy sets. As a result, this work presents a unique two-staged spherical fuzzy MCDM method for solving package tour selection problems in the Vietnamese tourism sector using SF-Delphi and SF-TOPSIS.By combining spherical fuzzy sets and the conventional Delphi and TOPSIS to evaluate the tour package selection process, this study makes the following contributions:(1)
This study provides a clear and comprehensive view of the critical determinants in selecting package tour products.(2)
This study is the first to develop the Delphi method based on spherical fuzzy sets. The spherical weighted geometric mean (SWGM) operator is presented to collect expert opinions and calculate the threshold for excluding less significant criteria using the spherical scoring function.(3)
This study pioneers to propose a novel two-staged SF-Delphi and SF-TOPSIS approach to assist tour operators and travel agencies in understanding how the customers select a package tour.(4)
The findings of this study contribute to developing marketing strategies and assisting tour operators in running an effective marketing program.The remainder of this study is divided into the following sections: Section2 is a review of the literature. Section 3 offers a methodological framework for developing the unique SF-Delphi and SF-TOPSIS systems. Section 4 reports an illustration example and a comparison with other decision-making methods. Section 5 summarizes the paper’s conclusion, limitations, and ramifications.


## 2. Literature Review

### 2.1. Literature Review on Package Tours

The field of package tours has been studied since early. Weightman [40] studied 13 packages tours in India to identify customers' experience toward natural traveling via only two components, including vehicles and hotels. The research then indicated that the difference between modern and traditional cities caused the classification of Indian tourists. Also, focusing on a package tour, Gratton and Richards [41] research the issue in the UK and Germany related to economic matters. The study distinguished the conduct and performance of package tours in these two markets; the signature difference is that the UK is very competitive while the German market is an oligopoly. This led to UK companies’ domestic concentration and Europe’s expansion of German ones. The research also indicated that UK package tours were influenced by price, profit margins, and the ease of immigration for the travel business.In contrast, Davies and Downward [42] also chose the behaviors and the exact topic of package tours, but the research objects were tourism companies in the UK. Also, focus on customers’ experience in package tours, but Tucker [43] chose to study the relationship between tourists’ tour production process and tour consumption behavior in the neighboring market of Australia. The paper revealed the role of natural and green landscapes of New Zealand in the consumption decision of the tourists and tourist experience was built by the negotiation process between tour operators and consumers, including the performative factors from the producers. Like Davies and Downward [42], Räikkönen and Honkanen [44] studied tour operators’ experience to examine tourists’ experience at the destination, starting from tour building and delivering to tourists. The result of the customer satisfaction evaluation showed that prearrival service influenced the tourist decision, although the destination service and accommodation services were the most critical elements. However, the research result of Räikkönen and Honkanen [44] was in contrast with Davies and Downward [42], which exposed that tourists’ satisfaction did not have an enormous influence on the success of a package tour experience as the figures were only 34% of the variance. The paper concluded that the tour experience was multifaceted and hybrid and affected by numerous factors and actors. From a different point of view, Cheng et al. [45] researched tour leaders’ role in the tourists’ decisions and their traveling behavior. The paper results revealed potential future discussion and further study regarding the role of tour leaders in package tours.Tucker [43] identified numerous elements influencing travelers’ choices, including destination amenities, tourism infrastructure, natural characteristics, human resources, and price. Similar research had been conducted for the Turkey market with Iranian tourists as research objects. However, the result was slightly different; Ozturkoglu et al. [46] claimed that entertainment, a family-friendly destination, the weather, cultural resources, and the quality of resort hotels were the primary variables that influenced tourists’ travel decisions. The study also indicated that the main reasons that tourists chose all-inclusive package tours were the preplan, service quality, and eliminated extra expenses. In recent research, Liao and Chuang [2] investigate the integration of package tour designs’ different attributes with the self-building experience of the Taiwanese tourists at the destination. The paper indicated that tourists prioritize attractions, lodging, duration of stay, pricing, food, transportation, and season. The research concluded that the travel agency should optimize resource usage to grow the international tourist industry through enhanced customer experience.


### 2.2. Literature Review on MCDM Applications

MCDM is a suitable model for this field as it allows for evaluating various factors so that the analysis results are proper based on multicriteria. In addition, the hospitality industry has a typical characteristic; it is the fuzzy and ambiguous service quality and customer satisfaction that requires a fuzzy researching method to investigate [21].The Delphi method [47] has been used to obtain a consensus of answers through questionnaires in many research areas. The Delphi method requires that some experts answer a series of questionnaires through several rounds. After each round, the facilitator asks each expert to refine his previous response based on other experts’ opinions. After several rounds, a census of the experts’ opinions is formed on the correct answers to the questionnaires. However, the Delphi method is relatively time-consuming due to performing several questionnaire rounds [48]. In addition to that, the experts’ judgments might be uncertain since there may exist ambiguity when experts answer the questionnaires due to the differences in the meanings and interpretations of the criteria. The original Delphi technique faced criticism for convergence, uncertainty, and vagueness of expert opinion initiated by the repetitive survey. To overcome the drawbacks of the Delphi model, the fuzzy set theory was integrated with the traditional Delphi technique to accomplish a decision made by a group of experts by addressing the fuzziness of the judgments [49, 50].Additionally, a hybrid Delphi-Fuzzy-TOPSIS approach has been used to clarify the importance of the determinants and the ideal placement [51]. The fuzzy-Delphi method combines the benefits of fuzzy theory and the Delphi technique. Even when a tiny sample size was used, the Delphi technique produced objective and realistic results [52]. As a result, the hybrid strategy significantly reduced the time and expense associated with achieving consensus without distorting the outcomes [53]. Furthermore, TOPSIS is frequently used to tackle multicriteria decision-making problems due to its success in prioritizing options and computational simplicity [54, 55]. However, the conventional TOPSIS technique with its crisp numerical values is incompatible with favored models [56]. As a result, the fuzzy TOPSIS approach was successfully created to improve the comprehensive and reasonable assessment of alternative performance when decision makers’ judgments and linguistic assessments are ambiguous and vague in a multicriteria decision setting [57, 58].Due to the instinctual nature of human thought, evaluations of alternatives are invariably imprecise and opaque in real-world decision-making situations. The requirement to cope with uncertainty in real-world problems has resulted in the development of numerous extensions of fuzzy sets [59]. First, Zadeh [60] proposed type-2 fuzzy sets to extend typical fuzzy sets for determining a fuzzy set’s exact membership function. This fuzzy set has an excessive number of parameters and thus cannot be used for problem modeling. Additionally, many scholars overlook the third dimension for the sake of simplicity. Atanassov and Gargov [61] suggested intuitionistic fuzzy sets, which treat an element’s membership and nonmembership degrees as independent elements but with the constraint that their sum is within the interval [0, 1]. Hesitant fuzzy sets [62] were introduced as a helpful tool for determining the membership degrees of numerous candidate items in a set. These fuzzy sets allow for the possibility of an element having multiple degrees of membership between zero and one. Yager and Abbasov [63] constructed Pythagorean fuzzy sets with a membership degree and a nonmembership degree that satisfy the condition that the square sum of the membership and nonmembership degrees equals one [64]. Ali and Smarandache [65] proposed neutrosophic fuzzy sets to generalize intuitionistic fuzzy sets. Cường [66] invented picture fuzzy sets. When confronted with human opinions involving many response types: yes, abstain, no, and rejection, picture fuzzy set-based models, may be acceptable. These sets enable decision-makers to assign membership, nonmembership, and reluctance degrees over a greater area. Spherical fuzzy sets, the latest extension of fuzzy sets, allow for the expression of an expert’s indeterminacy, membership, and non-membership degrees. Experts may assign any combination of the three characteristics to remain within the unit sphere. This amazing property distinguishes the spherical fuzzy sets from other fuzzy set models. The notion of spherical fuzzy sets is favorable and effective for dealing with uncertainty and imprecision in multiattribute decision-making issues [34].Zoraghi et al. [4] evaluated and ranked service quality in the hotel business using a fuzzy MCDM model in the tourism and hospitality industry. Tseng [67] used a combined fuzzy TOPSIS and the decision making trial and evaluation laboratory (DEMATEL) technique based on linguistic preference to investigate the service quality expectations in the hot spring hotel ranking problem. Guo and He [68] examined the collaboration issues between travel agencies and hotels operating luxury and economic package tours. However, in cases where proposed criteria were derived from a literature review, their study did not examine their validity. Apart from studies that use traditional fuzzy-MCDM techniques, no research has combined spherical fuzzy sets with MCDM in a Delphi survey. In the current context of tour package selection, the spherical fuzzy sets are deployed to overcome the limitations of the conventional Delphi and TOPSIS techniques under ambiguity and complexity. This is the first study in Vietnam that proposes the SF-Delphi and SF-TOPSIS to investigate the comprehensive criteria influencing customer decisions in package tour selection and prioritizing package tour operators.


### 2.3. Literature Review on Criteria

After an in-depth and comprehensive literature review, various criteria have been related to the customer’s decision-making process in selecting a package tour. This section presents some main critical criteria based on the literature review and experts’ opinions as shown in Table1.Price (PR): Price is always one determinant factor that significantly influences the customers’ decisions and is a must-considered component while building the marketing strategy [69]. However, a low-price tour package may positively affect the short-term period but negatively affect the long term. Besides the package price, Lin and Kuo [70] also proved that the cost of transportation and shopping also influences the customers’ decisions. The price of the tour even referred to the quality of the service [71].Service quality (SQ): The quality of the service provided is always the determinant factor that significantly influences the customers’ decisions [72]. Lin [73] proved that besides transportation mode, service quality was the primary indicator of tourists’ travel motivation for slow travel, becoming famous for sustainable tourism. Lin and Kuo [70] defined the hotel’s service quality as increasing accommodation willingness and reducing customer complaints. Studying service quality from different points of view, Cheng et al. [45] identified the importance of the tour leader role in the customers’ decision-making process. Supporting this idea, Chang [74] proved that tour-guide performance greatly influenced the tourists’ satisfaction and experience, especially shopping behavior, which can contribute directly to the company’s revenue [75].Information & Technology (IT): People have witnessed the vast development and application of technology in nearly all aspects of life in the last decades [76, 77]. As in the trend, technology is also applied in tourism and has transformed the industry and changed customers’ behavior [78, 79]. According to Wang and Fesenmaier [80], mobile technology has significantly changed tourists’ decision-making to opinion-based information collection based on prototypes. A survey to analyze the impact of technology on tourist behavior proved that the Internet was the most tourists’ source of accommodation, transportation, map-based, and attractions [81]. At various points of their decision-making process, tourists use mobile technologies. As a result, mobile technology is suitable for destination promotion in the travel and tourism industry [82]. Additionally, cellphones can drastically alter the travel experience by replacing traditional methods of obtaining information, picking a location, exploring that destination, and post-tour management [83, 84].Location (LO): Tourists may have different aims and reasons to choose a destination [85, 86] Tourism destinations could be classified into different categories to satisfy the different traveling purposes [87]. Destinations are always the priority when it comes to a decision, but the previous studies have not clarified between geographical and activities or experience destinations [88]. Defining factors contributing to the customers’ location decision will help marketers and travel companies find an effective marketing strategy and attractive service to yield revenue [89].Table 1
Criteria utilized in previous research.
CriteriaExplanationReferencesPrice (PR1)Tour cost (PR 1)Tour cost is understood as the cost of traveling insurance based on the standard of the travel agency, accommodation, daily meals (breakfast, lunch and dinner, sight-seeing tickets, transportation cost, at the destination within the package tour dates.[69]Transportation cost (PR 2)A selection of transportation costs will help enhance the customers' satisfaction because they have different budget planning.[70]Insurance included option (PR 3)Option to buy insurance included in the tour[71]Package discount (PR 4)A discount on package tour price will motivate customers to book the tour[70]Shopping expense (PR 5)Price and frequency of shopping arrangement at the destination[1]Service quality (SQ)Hotel quality (SQ 1)The hotel quality is evaluated by the star rating, the responsiveness and expertise of the staff, security, equipment, and ease of access. Included hotel service, hotel facilities, and hotel staff quality[70]F&B quality (SQ 2)Quality of food and beverage provided during tour time[72]Tour-guide performance (SQ 3)Tour guide performance significantly affects the customers' satisfaction and experience during the tour time and can lead to repurchase or word-of-mouth activities of the customers[73]Tour operator quality (SQ 4)Tourism and service organizations have fiercely competed to attract tourists and acquire market share and profit by improving service quality and marketing guidelines to persuade tourists to return to an area.[74]Tour consulting quality (SQ5)The quality of sales/consulting staff on providing service to the customers[75]Transportation quality (SQ 6)Diversity of transportation supply and the satisfaction of customers toward transportation quality significantly influence their experience[70]Travel company post-tour service (SQ 7)A tourist may engage in the future for the products and services they have consumed and experienced, tourist destinations they have visited, or even travel service enterprises, such as revisit, the recommendation to others, or word-of-mouth publicity for the products and services.[72]Information and technology (IT)Marketing technology (IT1)Marketing technology included all the modern information communication technology used for marketing and tour promotion to the customers: social network, social media, advertisement tools, and technical, analytic tools served for marketing purposes.[76, 77]Tour operator website quality (IT2)Reliable and trustful information on the website will encourage the customers to choose the destination and book the tour. Website content, information quality, correct information creates trust and the same experience and expectation.[78, 79]Information communication between customers (IT3)The information shared among travelers on social media or public websites may affect the customers' decision to visit a destination. It could be positive or negative depending on the experience of previous customers[80, 81]Payment method (IT4)It is necessary to study the influence of online payment methods on the customers' decision-making process.[82]Communication technology of tour company (IT 5)The technology used to communicate between tour consultants and tour operators, and customers before, during, and after tour shows the company's professionality and affects the customers’ experience and perception.[83, 84]Location (LO)Location environment & climate (LO1)A distinctive characteristic of the destination's environment and climate may directly influence the decision to purchase the tour of the customers.[85]Attractive spots, local leisure & recreation (LO2)Classification and diversity of attractive spots and leisure places can offer customers a different experience, and customers may choose the one that matches their traveling purpose.[86, 87]Local culture (LO3)Typical cultural characteristics could be an attractive point to encourage the customers to choose the destination.[88]Travel risks (LO4)Customers’ perception of the location’s risk of robbery, social security and safety of the location.[89]


## 2.1. Literature Review on Package Tours

The field of package tours has been studied since early. Weightman [40] studied 13 packages tours in India to identify customers' experience toward natural traveling via only two components, including vehicles and hotels. The research then indicated that the difference between modern and traditional cities caused the classification of Indian tourists. Also, focusing on a package tour, Gratton and Richards [41] research the issue in the UK and Germany related to economic matters. The study distinguished the conduct and performance of package tours in these two markets; the signature difference is that the UK is very competitive while the German market is an oligopoly. This led to UK companies’ domestic concentration and Europe’s expansion of German ones. The research also indicated that UK package tours were influenced by price, profit margins, and the ease of immigration for the travel business.In contrast, Davies and Downward [42] also chose the behaviors and the exact topic of package tours, but the research objects were tourism companies in the UK. Also, focus on customers’ experience in package tours, but Tucker [43] chose to study the relationship between tourists’ tour production process and tour consumption behavior in the neighboring market of Australia. The paper revealed the role of natural and green landscapes of New Zealand in the consumption decision of the tourists and tourist experience was built by the negotiation process between tour operators and consumers, including the performative factors from the producers. Like Davies and Downward [42], Räikkönen and Honkanen [44] studied tour operators’ experience to examine tourists’ experience at the destination, starting from tour building and delivering to tourists. The result of the customer satisfaction evaluation showed that prearrival service influenced the tourist decision, although the destination service and accommodation services were the most critical elements. However, the research result of Räikkönen and Honkanen [44] was in contrast with Davies and Downward [42], which exposed that tourists’ satisfaction did not have an enormous influence on the success of a package tour experience as the figures were only 34% of the variance. The paper concluded that the tour experience was multifaceted and hybrid and affected by numerous factors and actors. From a different point of view, Cheng et al. [45] researched tour leaders’ role in the tourists’ decisions and their traveling behavior. The paper results revealed potential future discussion and further study regarding the role of tour leaders in package tours.Tucker [43] identified numerous elements influencing travelers’ choices, including destination amenities, tourism infrastructure, natural characteristics, human resources, and price. Similar research had been conducted for the Turkey market with Iranian tourists as research objects. However, the result was slightly different; Ozturkoglu et al. [46] claimed that entertainment, a family-friendly destination, the weather, cultural resources, and the quality of resort hotels were the primary variables that influenced tourists’ travel decisions. The study also indicated that the main reasons that tourists chose all-inclusive package tours were the preplan, service quality, and eliminated extra expenses. In recent research, Liao and Chuang [2] investigate the integration of package tour designs’ different attributes with the self-building experience of the Taiwanese tourists at the destination. The paper indicated that tourists prioritize attractions, lodging, duration of stay, pricing, food, transportation, and season. The research concluded that the travel agency should optimize resource usage to grow the international tourist industry through enhanced customer experience.


## 2.2. Literature Review on MCDM Applications

MCDM is a suitable model for this field as it allows for evaluating various factors so that the analysis results are proper based on multicriteria. In addition, the hospitality industry has a typical characteristic; it is the fuzzy and ambiguous service quality and customer satisfaction that requires a fuzzy researching method to investigate [21].The Delphi method [47] has been used to obtain a consensus of answers through questionnaires in many research areas. The Delphi method requires that some experts answer a series of questionnaires through several rounds. After each round, the facilitator asks each expert to refine his previous response based on other experts’ opinions. After several rounds, a census of the experts’ opinions is formed on the correct answers to the questionnaires. However, the Delphi method is relatively time-consuming due to performing several questionnaire rounds [48]. In addition to that, the experts’ judgments might be uncertain since there may exist ambiguity when experts answer the questionnaires due to the differences in the meanings and interpretations of the criteria. The original Delphi technique faced criticism for convergence, uncertainty, and vagueness of expert opinion initiated by the repetitive survey. To overcome the drawbacks of the Delphi model, the fuzzy set theory was integrated with the traditional Delphi technique to accomplish a decision made by a group of experts by addressing the fuzziness of the judgments [49, 50].Additionally, a hybrid Delphi-Fuzzy-TOPSIS approach has been used to clarify the importance of the determinants and the ideal placement [51]. The fuzzy-Delphi method combines the benefits of fuzzy theory and the Delphi technique. Even when a tiny sample size was used, the Delphi technique produced objective and realistic results [52]. As a result, the hybrid strategy significantly reduced the time and expense associated with achieving consensus without distorting the outcomes [53]. Furthermore, TOPSIS is frequently used to tackle multicriteria decision-making problems due to its success in prioritizing options and computational simplicity [54, 55]. However, the conventional TOPSIS technique with its crisp numerical values is incompatible with favored models [56]. As a result, the fuzzy TOPSIS approach was successfully created to improve the comprehensive and reasonable assessment of alternative performance when decision makers’ judgments and linguistic assessments are ambiguous and vague in a multicriteria decision setting [57, 58].Due to the instinctual nature of human thought, evaluations of alternatives are invariably imprecise and opaque in real-world decision-making situations. The requirement to cope with uncertainty in real-world problems has resulted in the development of numerous extensions of fuzzy sets [59]. First, Zadeh [60] proposed type-2 fuzzy sets to extend typical fuzzy sets for determining a fuzzy set’s exact membership function. This fuzzy set has an excessive number of parameters and thus cannot be used for problem modeling. Additionally, many scholars overlook the third dimension for the sake of simplicity. Atanassov and Gargov [61] suggested intuitionistic fuzzy sets, which treat an element’s membership and nonmembership degrees as independent elements but with the constraint that their sum is within the interval [0, 1]. Hesitant fuzzy sets [62] were introduced as a helpful tool for determining the membership degrees of numerous candidate items in a set. These fuzzy sets allow for the possibility of an element having multiple degrees of membership between zero and one. Yager and Abbasov [63] constructed Pythagorean fuzzy sets with a membership degree and a nonmembership degree that satisfy the condition that the square sum of the membership and nonmembership degrees equals one [64]. Ali and Smarandache [65] proposed neutrosophic fuzzy sets to generalize intuitionistic fuzzy sets. Cường [66] invented picture fuzzy sets. When confronted with human opinions involving many response types: yes, abstain, no, and rejection, picture fuzzy set-based models, may be acceptable. These sets enable decision-makers to assign membership, nonmembership, and reluctance degrees over a greater area. Spherical fuzzy sets, the latest extension of fuzzy sets, allow for the expression of an expert’s indeterminacy, membership, and non-membership degrees. Experts may assign any combination of the three characteristics to remain within the unit sphere. This amazing property distinguishes the spherical fuzzy sets from other fuzzy set models. The notion of spherical fuzzy sets is favorable and effective for dealing with uncertainty and imprecision in multiattribute decision-making issues [34].Zoraghi et al. [4] evaluated and ranked service quality in the hotel business using a fuzzy MCDM model in the tourism and hospitality industry. Tseng [67] used a combined fuzzy TOPSIS and the decision making trial and evaluation laboratory (DEMATEL) technique based on linguistic preference to investigate the service quality expectations in the hot spring hotel ranking problem. Guo and He [68] examined the collaboration issues between travel agencies and hotels operating luxury and economic package tours. However, in cases where proposed criteria were derived from a literature review, their study did not examine their validity. Apart from studies that use traditional fuzzy-MCDM techniques, no research has combined spherical fuzzy sets with MCDM in a Delphi survey. In the current context of tour package selection, the spherical fuzzy sets are deployed to overcome the limitations of the conventional Delphi and TOPSIS techniques under ambiguity and complexity. This is the first study in Vietnam that proposes the SF-Delphi and SF-TOPSIS to investigate the comprehensive criteria influencing customer decisions in package tour selection and prioritizing package tour operators.


## 2.3. Literature Review on Criteria

After an in-depth and comprehensive literature review, various criteria have been related to the customer’s decision-making process in selecting a package tour. This section presents some main critical criteria based on the literature review and experts’ opinions as shown in Table1.Price (PR): Price is always one determinant factor that significantly influences the customers’ decisions and is a must-considered component while building the marketing strategy [69]. However, a low-price tour package may positively affect the short-term period but negatively affect the long term. Besides the package price, Lin and Kuo [70] also proved that the cost of transportation and shopping also influences the customers’ decisions. The price of the tour even referred to the quality of the service [71].Service quality (SQ): The quality of the service provided is always the determinant factor that significantly influences the customers’ decisions [72]. Lin [73] proved that besides transportation mode, service quality was the primary indicator of tourists’ travel motivation for slow travel, becoming famous for sustainable tourism. Lin and Kuo [70] defined the hotel’s service quality as increasing accommodation willingness and reducing customer complaints. Studying service quality from different points of view, Cheng et al. [45] identified the importance of the tour leader role in the customers’ decision-making process. Supporting this idea, Chang [74] proved that tour-guide performance greatly influenced the tourists’ satisfaction and experience, especially shopping behavior, which can contribute directly to the company’s revenue [75].Information & Technology (IT): People have witnessed the vast development and application of technology in nearly all aspects of life in the last decades [76, 77]. As in the trend, technology is also applied in tourism and has transformed the industry and changed customers’ behavior [78, 79]. According to Wang and Fesenmaier [80], mobile technology has significantly changed tourists’ decision-making to opinion-based information collection based on prototypes. A survey to analyze the impact of technology on tourist behavior proved that the Internet was the most tourists’ source of accommodation, transportation, map-based, and attractions [81]. At various points of their decision-making process, tourists use mobile technologies. As a result, mobile technology is suitable for destination promotion in the travel and tourism industry [82]. Additionally, cellphones can drastically alter the travel experience by replacing traditional methods of obtaining information, picking a location, exploring that destination, and post-tour management [83, 84].Location (LO): Tourists may have different aims and reasons to choose a destination [85, 86] Tourism destinations could be classified into different categories to satisfy the different traveling purposes [87]. Destinations are always the priority when it comes to a decision, but the previous studies have not clarified between geographical and activities or experience destinations [88]. Defining factors contributing to the customers’ location decision will help marketers and travel companies find an effective marketing strategy and attractive service to yield revenue [89].Table 1
Criteria utilized in previous research.
CriteriaExplanationReferencesPrice (PR1)Tour cost (PR 1)Tour cost is understood as the cost of traveling insurance based on the standard of the travel agency, accommodation, daily meals (breakfast, lunch and dinner, sight-seeing tickets, transportation cost, at the destination within the package tour dates.[69]Transportation cost (PR 2)A selection of transportation costs will help enhance the customers' satisfaction because they have different budget planning.[70]Insurance included option (PR 3)Option to buy insurance included in the tour[71]Package discount (PR 4)A discount on package tour price will motivate customers to book the tour[70]Shopping expense (PR 5)Price and frequency of shopping arrangement at the destination[1]Service quality (SQ)Hotel quality (SQ 1)The hotel quality is evaluated by the star rating, the responsiveness and expertise of the staff, security, equipment, and ease of access. Included hotel service, hotel facilities, and hotel staff quality[70]F&B quality (SQ 2)Quality of food and beverage provided during tour time[72]Tour-guide performance (SQ 3)Tour guide performance significantly affects the customers' satisfaction and experience during the tour time and can lead to repurchase or word-of-mouth activities of the customers[73]Tour operator quality (SQ 4)Tourism and service organizations have fiercely competed to attract tourists and acquire market share and profit by improving service quality and marketing guidelines to persuade tourists to return to an area.[74]Tour consulting quality (SQ5)The quality of sales/consulting staff on providing service to the customers[75]Transportation quality (SQ 6)Diversity of transportation supply and the satisfaction of customers toward transportation quality significantly influence their experience[70]Travel company post-tour service (SQ 7)A tourist may engage in the future for the products and services they have consumed and experienced, tourist destinations they have visited, or even travel service enterprises, such as revisit, the recommendation to others, or word-of-mouth publicity for the products and services.[72]Information and technology (IT)Marketing technology (IT1)Marketing technology included all the modern information communication technology used for marketing and tour promotion to the customers: social network, social media, advertisement tools, and technical, analytic tools served for marketing purposes.[76, 77]Tour operator website quality (IT2)Reliable and trustful information on the website will encourage the customers to choose the destination and book the tour. Website content, information quality, correct information creates trust and the same experience and expectation.[78, 79]Information communication between customers (IT3)The information shared among travelers on social media or public websites may affect the customers' decision to visit a destination. It could be positive or negative depending on the experience of previous customers[80, 81]Payment method (IT4)It is necessary to study the influence of online payment methods on the customers' decision-making process.[82]Communication technology of tour company (IT 5)The technology used to communicate between tour consultants and tour operators, and customers before, during, and after tour shows the company's professionality and affects the customers’ experience and perception.[83, 84]Location (LO)Location environment & climate (LO1)A distinctive characteristic of the destination's environment and climate may directly influence the decision to purchase the tour of the customers.[85]Attractive spots, local leisure & recreation (LO2)Classification and diversity of attractive spots and leisure places can offer customers a different experience, and customers may choose the one that matches their traveling purpose.[86, 87]Local culture (LO3)Typical cultural characteristics could be an attractive point to encourage the customers to choose the destination.[88]Travel risks (LO4)Customers’ perception of the location’s risk of robbery, social security and safety of the location.[89]


## 3. Methodology

### 3.1. Spherical Fuzzy Sets: Basic Operations and Definitions

Kutlu Gündoğdu and Kahraman [34] proposed spherical fuzzy sets (SFS) as the most recent development of fuzzy sets, with each spherical fuzzy number representing the membership, non-membership, and hesitation functions in the interval [0, 1] (Figure 1).Figure 1
Geometric representations of SFS.Definition 1.
SFS is presented asF˜S:(1)F˜S=x,αF˜Sx,βF˜Sx,γF˜Sx|x∈X,where F˜S denotes a spherical fuzzy set of the universe X.(2)αF˜Sx:X⟶0,1,βF˜Sx:X⟶0,1,γF˜Sx:X⟶0,1,(3)0≤αF˜S2x+βF˜S2x+γF˜S2x≤1,with ∀x∈X, for each x, αF˜Sx,βF˜Sx, and γF˜Sx denote for membership, non-membership, and hesitancy levels of x to A˜S, respectively.Definition 2.
LetF˜S=αF˜S,βF˜S,γF˜S and E˜S=αE˜S,βE˜S,γE˜S be two SFSs. Some arithmetic operations of SFS are presented as follows.(i)
Union(4)F˜S∪E˜S=maxαF˜S,αE˜S,minβF˜S,βE˜S,min1−maxcv2+minβF˜S,βE˜S20.5,maxγF˜S,γE˜S.(ii)
Intersection(5)F˜S∩E˜S=minαF˜S,αE˜S,maxβF˜S,βE˜S,max1−minαF˜S,αE˜S2+maxβF˜S,βE˜S21/2,minγF˜S,γE˜S.(iii)
Addition(6)F˜S⊕E˜S=αF˜S2+αE˜S2−αF˜S2αE˜S20.5,βF˜SβE˜S,1−αE˜S2γF˜S2+1−αF˜S2γE˜S2−γF˜S2γE˜S20.5(iv)
Multiplication(7)F˜S⊕E˜S=αF˜SβF˜S,βF˜S2+βE˜S2−βF˜S2βE˜S20.5,1−βE˜S2γF˜S2+1−βF˜S2γE˜S2−γF˜S2γE˜S20.5.(v)
Multiplication by a scalar;σ>0(8)σ.F˜S=1−1−αF˜S2σ0.5,βF˜Sσ,1−αF˜S2σ−1−αF˜S2−γF˜S2σ0.5.(vi)
Power ofFS;σ>0(9)F˜Sσ=αF˜Sσ,1−1−βF˜S2σ0.5,1−βF˜S2σ−1−βFS2−γF˜S2σ0.5.Definition 3.
For these SFSsF˜S=αF˜S,βF˜S,γF˜S and E˜S=αE˜S,βE˜S,γE˜S, the followings are valid under the condition σ,σ1,σ2>0.(10)F˜S⊕E˜S=E˜S⊕F˜S,F˜S⊗E˜S=E˜S⊗F˜S,σF˜S⊕E˜S=σF˜S⊕σE˜S,σ1F˜S⊕σ2F˜S=σ1+σ2F˜S,F˜S⊗E˜Sσ=F˜Sσ⊗E˜Sσ,F˜Sσ1⊗F˜Sσ2=F˜Sσ1+σ2.Definition 4.
Spherical weighted arithmetic mean (SWAM) concerningw=w1,w2,…,wn; wi∈0,1; ∑i=1nwi=1, SWAM is defined as follows.(11)SWAMwF˜S1,…,F˜Sn=w1F˜S1+w2F˜S2+…+wnF˜Sn=1−∏i=1n1−αF˜Si2wi0.5,∏i=1nβF˜Siwi,∏i=1n1−αF˜Si2wi−∏i=1n1−αF˜Si2−γF˜Si2wi0.5.Definition 5.
Spherical weighted geometric mean (SWGM) concerningw=w1,w2…,wn; wi∈0,1; ∑i=1nwi=1, SWGM is defined as follows.(12)SWGMwF˜S1,…,F¨Sn=F˜S1w1+F˜S2w2+…+F˜Snwn=∏i=1nαF˜Siwi,1−∏i=1n1−βF˜Si2wi0.5,∏i=1n1−βF˜Si2−∏i=1n1−βF˜Si2−γF˜Si2wi0.5.


### 3.2. Proposed Method

This research aims to propose the novel two-phased spherical fuzzy MCDM approach of SF-Delphi and SF-TOPSIS (Figure2):Phase I: In this study, a novel modified Delphi method incorporating spherical fuzzy sets (SF-Delphi) is proposed to reduce the number of responses and investigation time required for an adequate assessment and transformation of the spherical fuzzy evaluation into accurate data from step one to step three. A preliminary set of tour package criteria was distributed to a panel of specialists, ranging from tourism and hospitality executives to several academics, who used a spherical fuzzy scale to rank the importance of each criterion based on their professional knowledge. In phase I, the novel modified SF-Delphi method allows the obtaining of a consensus evaluation from the panelists by applying the first three steps as follows:Step 1: Experts’ opinions are aggregated and assessed. The respondents were asked to rate the criteria using the linguistic terms shown in Table2. The significance vector for each indicator is obtained using the SWGM operator [90] and is shown in equation below:(13)F˜agg=SWGMwF˜S1,…,F¨Sn=F˜S1w1+F˜S2w2+…+F˜Snwn=∏i=1nαF˜Siwi,1−∏i=1n1−βF˜Si2wi0.5,∏i=1n1−βF˜Si2wi−∏i=1n1−βF˜Si2−γF˜Si2wi0.5(14)F˜agg=α11,β11,γ11⋯α1m,β1m,γ1m⋮⋱⋮αn1,βn1,γn1⋯αnm,βnm,γnm.Step 2: Differentiating from the previous Delphi method [52, 91], we propose to defuzzy the aggregated criteria score using (15), the score function.(15)Scoredi=2αij−γij2−βij−γij2.Afterward, the threshold is attained asDi=∑i=1ndi/m to validate criteria from the initial set. If di>D, criterion Ci is valid, and if di<D, criterion Ci is removed.Step 3: We employ the SWAM operator’s subjective weighting method extension of the spherical fuzzy sets in this study. Obtaining weights by Equations (14-15).Phase II: The SF-TOPSIS model was proposed by [90] to rank the alternative regarding proposed criteria from step 4 to step 10.Step 4: In this study, we aggregate the judgments of each decision-maker (DM) using the SWAM and SWGM operators from Definitions1 and 5. Construct aggregated spherical fuzzy decision matrix based on decision-makers’ opinions :(16)D˜=CjX˜imxn=α11,β11,γ11α11,β11,γ11⋯α1n,β1n,γ1nα21,β21,γ21α22,β22,γ22⋯α2n,β2n,γ2n⋮⋮⋮αm1,βm1,γm1αm2,βm2,γm2⋯αmn,βmn,γmnStep 5: Aggregating the spherical fuzzy linguistic evaluations of the decision criteria assigned by decision-makers. The first possible way is to follow a partially fuzzy approach: Defuzzify the aggregated criteria weights by using the score function given in equation (15). Normalize the aggregated criteria weights by using (17)w¯sj=wjs∑i=1nwjs.The second way of the complete fuzzy approach is to continue without defuzzifying the criteria weights.Step 6: Using the score values acquired in Step 6, determine the spherical fuzzy positive ideal solution (SF-PIS) and the spherical fuzzy negative ideal solution (SF–NIS). Equation (18) is used to get the maximum scores in the decision matrix for the SF-PIS. The associated SFN numbers are determined using the crisp maximum scores as in equation (19).(18)X∗=Cj,maxi<ScoreCjX˜i>|j=1,2,…,n,(19)X˜∗=C1,α1∗,β1∗,γ1∗,C2,α2∗,β2∗,γ2∗…Cn,αn∗,βn∗,γn∗.For the SF-NIS, (20) is used to find the minimum scores in the decision matrix. The corresponding SF numbers are determined based on the crisp minimum scores as in (21).(20)X−=Cj,mini<ScoreCjX˜i>|j=1,2,…,n,(21)X˜−=C1,α1−,β1−,γ1−,C2,α2−,β2−,γ2−…Cn,αn−,βn−,γn−.Step 7: Calculating the distances between alternativeXi. The SF-NIS and SF-PIS are calculated using equation (22-23), respectively.(22)DXi,X−=12n∑i=1nαxi−αx−2+βxi−βx−2+γxi−γx−2,(23)DXi,X∗=12n∑i=1nαxi−αx∗2+βxi−βx∗2+γxi−γx∗2.Step 8: Determining the maximum distance to the SF-NIS and minimum distance to the SF-PIS using equations (24) and (25), respectively.(24)DmaxXi,X−=max1≤i≤mDXi,X−,(25)DminXi,X∗=min1≤i≤mDXi,X∗.Step 9: The revised closeness ratio is calculated. Equation (26) results in zero or negative values, as the second element in the subtraction is at least equal to the first. We modified this equivalence as Equation (27) to produce zero or positive outcomes.(26)ClosenessRatioXi=ξXi=DXi,X−DmaxXi,X−−DXi,X∗DminXi,X∗,(27)ClosenessRatioXi=ξXi=DXi,X∗DminXi,X∗−DXi,X−DmaxXi,X−.Step 10: Equation (27) is used in this study to calculate the alternatives’ ideal ranking order and select the optimal alternative to the increasing values of the closeness ratio.Figure 2
Proposed research framework.Table 2
Spherical linguistic expressions.
Linguistic scaleCodeα,β,γAbsolutely more importanceAMI(0.9, 0.1, 0.1)Very high importanceVHI(0.8, 0.2, 0.2)High importanceHI(0.7, 0.3, 0.3)Slightly more importanceSMI(0.6, 0.4, 0.4)Equally importanceEI(0.5, 0.5, 0.5)Slightly low importanceSLI(0.4, 0.6, 0.4)Low importanceLI(0.3, 0.7, 0.3)Very low importanceVLI(0.2, 0.8, 0.2)Absolutely low importanceALI(0.1, 0.9, 0.1)


## 3.1. Spherical Fuzzy Sets: Basic Operations and Definitions

Kutlu Gündoğdu and Kahraman [34] proposed spherical fuzzy sets (SFS) as the most recent development of fuzzy sets, with each spherical fuzzy number representing the membership, non-membership, and hesitation functions in the interval [0, 1] (Figure 1).Figure 1
Geometric representations of SFS.Definition 1.
SFS is presented asF˜S:(1)F˜S=x,αF˜Sx,βF˜Sx,γF˜Sx|x∈X,where F˜S denotes a spherical fuzzy set of the universe X.(2)αF˜Sx:X⟶0,1,βF˜Sx:X⟶0,1,γF˜Sx:X⟶0,1,(3)0≤αF˜S2x+βF˜S2x+γF˜S2x≤1,with ∀x∈X, for each x, αF˜Sx,βF˜Sx, and γF˜Sx denote for membership, non-membership, and hesitancy levels of x to A˜S, respectively.Definition 2.
LetF˜S=αF˜S,βF˜S,γF˜S and E˜S=αE˜S,βE˜S,γE˜S be two SFSs. Some arithmetic operations of SFS are presented as follows.(i)
Union(4)F˜S∪E˜S=maxαF˜S,αE˜S,minβF˜S,βE˜S,min1−maxcv2+minβF˜S,βE˜S20.5,maxγF˜S,γE˜S.(ii)
Intersection(5)F˜S∩E˜S=minαF˜S,αE˜S,maxβF˜S,βE˜S,max1−minαF˜S,αE˜S2+maxβF˜S,βE˜S21/2,minγF˜S,γE˜S.(iii)
Addition(6)F˜S⊕E˜S=αF˜S2+αE˜S2−αF˜S2αE˜S20.5,βF˜SβE˜S,1−αE˜S2γF˜S2+1−αF˜S2γE˜S2−γF˜S2γE˜S20.5(iv)
Multiplication(7)F˜S⊕E˜S=αF˜SβF˜S,βF˜S2+βE˜S2−βF˜S2βE˜S20.5,1−βE˜S2γF˜S2+1−βF˜S2γE˜S2−γF˜S2γE˜S20.5.(v)
Multiplication by a scalar;σ>0(8)σ.F˜S=1−1−αF˜S2σ0.5,βF˜Sσ,1−αF˜S2σ−1−αF˜S2−γF˜S2σ0.5.(vi)
Power ofFS;σ>0(9)F˜Sσ=αF˜Sσ,1−1−βF˜S2σ0.5,1−βF˜S2σ−1−βFS2−γF˜S2σ0.5.Definition 3.
For these SFSsF˜S=αF˜S,βF˜S,γF˜S and E˜S=αE˜S,βE˜S,γE˜S, the followings are valid under the condition σ,σ1,σ2>0.(10)F˜S⊕E˜S=E˜S⊕F˜S,F˜S⊗E˜S=E˜S⊗F˜S,σF˜S⊕E˜S=σF˜S⊕σE˜S,σ1F˜S⊕σ2F˜S=σ1+σ2F˜S,F˜S⊗E˜Sσ=F˜Sσ⊗E˜Sσ,F˜Sσ1⊗F˜Sσ2=F˜Sσ1+σ2.Definition 4.
Spherical weighted arithmetic mean (SWAM) concerningw=w1,w2,…,wn; wi∈0,1; ∑i=1nwi=1, SWAM is defined as follows.(11)SWAMwF˜S1,…,F˜Sn=w1F˜S1+w2F˜S2+…+wnF˜Sn=1−∏i=1n1−αF˜Si2wi0.5,∏i=1nβF˜Siwi,∏i=1n1−αF˜Si2wi−∏i=1n1−αF˜Si2−γF˜Si2wi0.5.Definition 5.
Spherical weighted geometric mean (SWGM) concerningw=w1,w2…,wn; wi∈0,1; ∑i=1nwi=1, SWGM is defined as follows.(12)SWGMwF˜S1,…,F¨Sn=F˜S1w1+F˜S2w2+…+F˜Snwn=∏i=1nαF˜Siwi,1−∏i=1n1−βF˜Si2wi0.5,∏i=1n1−βF˜Si2−∏i=1n1−βF˜Si2−γF˜Si2wi0.5.


## 3.2. Proposed Method

This research aims to propose the novel two-phased spherical fuzzy MCDM approach of SF-Delphi and SF-TOPSIS (Figure2):Phase I: In this study, a novel modified Delphi method incorporating spherical fuzzy sets (SF-Delphi) is proposed to reduce the number of responses and investigation time required for an adequate assessment and transformation of the spherical fuzzy evaluation into accurate data from step one to step three. A preliminary set of tour package criteria was distributed to a panel of specialists, ranging from tourism and hospitality executives to several academics, who used a spherical fuzzy scale to rank the importance of each criterion based on their professional knowledge. In phase I, the novel modified SF-Delphi method allows the obtaining of a consensus evaluation from the panelists by applying the first three steps as follows:Step 1: Experts’ opinions are aggregated and assessed. The respondents were asked to rate the criteria using the linguistic terms shown in Table2. The significance vector for each indicator is obtained using the SWGM operator [90] and is shown in equation below:(13)F˜agg=SWGMwF˜S1,…,F¨Sn=F˜S1w1+F˜S2w2+…+F˜Snwn=∏i=1nαF˜Siwi,1−∏i=1n1−βF˜Si2wi0.5,∏i=1n1−βF˜Si2wi−∏i=1n1−βF˜Si2−γF˜Si2wi0.5(14)F˜agg=α11,β11,γ11⋯α1m,β1m,γ1m⋮⋱⋮αn1,βn1,γn1⋯αnm,βnm,γnm.Step 2: Differentiating from the previous Delphi method [52, 91], we propose to defuzzy the aggregated criteria score using (15), the score function.(15)Scoredi=2αij−γij2−βij−γij2.Afterward, the threshold is attained asDi=∑i=1ndi/m to validate criteria from the initial set. If di>D, criterion Ci is valid, and if di<D, criterion Ci is removed.Step 3: We employ the SWAM operator’s subjective weighting method extension of the spherical fuzzy sets in this study. Obtaining weights by Equations (14-15).Phase II: The SF-TOPSIS model was proposed by [90] to rank the alternative regarding proposed criteria from step 4 to step 10.Step 4: In this study, we aggregate the judgments of each decision-maker (DM) using the SWAM and SWGM operators from Definitions1 and 5. Construct aggregated spherical fuzzy decision matrix based on decision-makers’ opinions :(16)D˜=CjX˜imxn=α11,β11,γ11α11,β11,γ11⋯α1n,β1n,γ1nα21,β21,γ21α22,β22,γ22⋯α2n,β2n,γ2n⋮⋮⋮αm1,βm1,γm1αm2,βm2,γm2⋯αmn,βmn,γmnStep 5: Aggregating the spherical fuzzy linguistic evaluations of the decision criteria assigned by decision-makers. The first possible way is to follow a partially fuzzy approach: Defuzzify the aggregated criteria weights by using the score function given in equation (15). Normalize the aggregated criteria weights by using (17)w¯sj=wjs∑i=1nwjs.The second way of the complete fuzzy approach is to continue without defuzzifying the criteria weights.Step 6: Using the score values acquired in Step 6, determine the spherical fuzzy positive ideal solution (SF-PIS) and the spherical fuzzy negative ideal solution (SF–NIS). Equation (18) is used to get the maximum scores in the decision matrix for the SF-PIS. The associated SFN numbers are determined using the crisp maximum scores as in equation (19).(18)X∗=Cj,maxi<ScoreCjX˜i>|j=1,2,…,n,(19)X˜∗=C1,α1∗,β1∗,γ1∗,C2,α2∗,β2∗,γ2∗…Cn,αn∗,βn∗,γn∗.For the SF-NIS, (20) is used to find the minimum scores in the decision matrix. The corresponding SF numbers are determined based on the crisp minimum scores as in (21).(20)X−=Cj,mini<ScoreCjX˜i>|j=1,2,…,n,(21)X˜−=C1,α1−,β1−,γ1−,C2,α2−,β2−,γ2−…Cn,αn−,βn−,γn−.Step 7: Calculating the distances between alternativeXi. The SF-NIS and SF-PIS are calculated using equation (22-23), respectively.(22)DXi,X−=12n∑i=1nαxi−αx−2+βxi−βx−2+γxi−γx−2,(23)DXi,X∗=12n∑i=1nαxi−αx∗2+βxi−βx∗2+γxi−γx∗2.Step 8: Determining the maximum distance to the SF-NIS and minimum distance to the SF-PIS using equations (24) and (25), respectively.(24)DmaxXi,X−=max1≤i≤mDXi,X−,(25)DminXi,X∗=min1≤i≤mDXi,X∗.Step 9: The revised closeness ratio is calculated. Equation (26) results in zero or negative values, as the second element in the subtraction is at least equal to the first. We modified this equivalence as Equation (27) to produce zero or positive outcomes.(26)ClosenessRatioXi=ξXi=DXi,X−DmaxXi,X−−DXi,X∗DminXi,X∗,(27)ClosenessRatioXi=ξXi=DXi,X∗DminXi,X∗−DXi,X−DmaxXi,X−.Step 10: Equation (27) is used in this study to calculate the alternatives’ ideal ranking order and select the optimal alternative to the increasing values of the closeness ratio.Figure 2
Proposed research framework.Table 2
Spherical linguistic expressions.
Linguistic scaleCodeα,β,γAbsolutely more importanceAMI(0.9, 0.1, 0.1)Very high importanceVHI(0.8, 0.2, 0.2)High importanceHI(0.7, 0.3, 0.3)Slightly more importanceSMI(0.6, 0.4, 0.4)Equally importanceEI(0.5, 0.5, 0.5)Slightly low importanceSLI(0.4, 0.6, 0.4)Low importanceLI(0.3, 0.7, 0.3)Very low importanceVLI(0.2, 0.8, 0.2)Absolutely low importanceALI(0.1, 0.9, 0.1)


## 4. Case Study

### 4.1. Description of a Case Study from Vietnam

This paper conducts a case study of Vietnam’s top ten package travel providers to validate the suggested model. Following the first evaluation, a panel of specialists (Table3) conducted surveys to determine the top 10 travel and tourist companies in 2019, including X1-Hanoitourist; X2-Pegas Tourist; X3-Vietravel; X4-TransViet Travel; X5-TST Tourist; X6-Anex Tour; X7-SaigonTourist; X8-BenThanh Tourist; X9-Datviet Tour; and X10-Exo Travel. The hierarchy of 21 critical criteria influencing a customer’s decision-making process in selecting a package tour is presented in Figure 3. The first phase deals with the critical criteria influencing customer’s decision-making process in selecting a package tour available in the literature, which is followed by the selections of the relevant dimensions (price, service quality, information and technology, and location) through the experts’ opinions with spherical fuzzy scales using SF-Delphi approach. Critical criteria relevant to package tour evaluation are investigated using the SF-TOPSIS approach to find the optimal providers in the second phase.Table 3
Expert profiles.
ExpertPositionGenderQualificationExperienceDM 1ManagerMaleMBA12DM 2ManagerFemaleMBA15DM 3ManagerMalePh.D19DM 4ManagerFemaleMBA18DM 5ManagerMaleMBA12DM 6ManagerFemalePh.D16DM 7ManagerMaleMBA13DM 8ManagerMaleMBA10DM 9Full ProfessorFemalePh.D10DM 10Distinguished ProfessorMalePost-Doctoral10DM 11Associate ProfessorMalePost-Doctoral15DM 12Associate ProfessorMalePh.D20Figure 3
Hierarchy of proposed criteria and alternatives.


### 4.2. Results of the SF-Delphi Method

The list of 21 possible indicators was compiled using secondary sources and expert consultants. The inquiry was divided into two sections and planned to be finished in 30 minutes. Section1 is devoted to the participant's demographic information. There were questions on the industry sector, a position occupied, education, and years of experience. They were then asked to express their degree of agreement with selected criteria based on their experience and expertise in the second section. To begin, invitations were sent through e-mail, and the questionnaire was sent only after approval. This study created an online questionnaire in English and Vietnamese using Google forms. The data collection process was active for three months, from May to August 2021. Twenty-six experts were contacted from various companies and academia for data collection. Out of the twenty-six experts, 12 experts agreed to be part of this study. Eight experts belong to different travel and tourism companies with more than ten years’ experience.Additionally, four experts have indulged in the research and teaching on tourism and hospitality industry for more than ten years. According to the recommendation of [92], 10 to 18 experts are to assure consensus among the participants. Table 3 summarizes the profiles of the experts. Table 4 shows how each expert described the significance of each critical criterion using spherical linguistic terms as defined in Table 2.Table 4
Spherical linguistic evaluation of criteria.
DimensionsCriteriaDM1DM2DM3DM4DM5DM6DM7DM8DM9DM10DM11DM12PricePR1AMIVHIVHIHIVHIHIVHIAMIHISMIVHIHIPR2VHIAMISMIHIAMIHISMIHISMIVHIAMIVHIPR3SMIHIAMIVHIHIVHIAMIVHIVHIHISMIHIPR4HIHIHIAMIVHIVHIHIHIVHIHIHIAMIPR5EISMIEISMISMIEIEISMISMIEIEISMIService qualitySQ1HIHIVHISMIAMISMIAMISMIHIVHIAMIVHISQ2VHISMIHIVHISMIAMIAMIHIHIVHIHIAMISQ3AMIVHISMIAMISMIVHIHIVHIHIVHIVHISMISQ4AMISMIHISMIAMIVHIAMIHIHIVHIAMIHISQ5SMIEIEISMISMIEIEISMIEIEISMIEISQ6VHIAMIVHIHIAMIHIVHIHIVHIHIHISMISQ7EISMISMIEIEISMISMIEISMISMIEISMIInformation and technologyIT1VHIHIAMIHIAMIHIVHIHISMIVHIVHISMIIT2HIVHIHIHIHIAMIHIHIVHIVHIVHIVHIIT3EISMISMISMIEIEISMISMIEIEISMISMIIT4AMIHIVHIVHIHISMIHIAMIHIHIVHIHIIT5VHIAMIEIAMIVHIAMIHIVHISMIVHIEIAMILocationLO1HIAMIHIVHIAMIVHIVHISMIHIHISMIVHILO2AMIVHIVHIHIVHIVHIEIAMIAMISMIVHIHILO3VHIHIVHIAMIVHIVHIAMIHIAMISMIVHIEILO4SMISMIAMIVHIAMIHIVHIHIVHIVHIHIVHIThen, experts’ opinions were converted into spherical fuzzy scales and aggregated using the SWGM operator in equation (13)-(14). Finally, spherical score values were defuzzified by (15). To eliminate the less important criteria, threshold (Di) = 1.2947. The detailed results of the SF-Delphi technique are shown in Table 5. Based on the comparisons between score value Sdi of each criterion and Threshold (Di), 4 criteria, including PR5; SQ5; SQ7; IT3 were rejected, and 17 critical criteria were accepted and are visualized in Figure 4.Table 5
Spherical fuzzy Delphi results.
CriteriaSFNsα,β,γScore valuesSdiDecisionsPR1(0.762, 0.250, 0.255)1.6102AcceptedPR2(0.742, 0.277, 0.285)1.4356AcceptedPR3(0.744, 0.270, 0.276)1.4677AcceptedPR4(0.755, 0.254, 0.257)1.5693AcceptedPR5(0.548, 0.454, 0.459)0.4051UnacceptedSQ1(0.742, 0.277, 0.285)1.4356AcceptedSQ2(0.751, 0.266, 0.272)1.5130AcceptedSQ3(0.743, 0.274, 0.282)1.4483AcceptedSQ4(0.759, 0.261, 0.268)1.5594AcceptedSQ5(0.539, 0.462, 0.467)0.3747UnacceptedSQ6(0.753, 0.258, 0.263)1.5477AcceptedSQ7(0.556, 0.446, 0.451)0.4377UnacceptedIT1(0.744, 0.270, 0.276)1.4677AcceptedIT2(0.756, 0.251, 0.253)1.5841AcceptedIT3(0.556, 0.446, 0.451)0.4377UnacceptedIT4(0.745, 0.266, 0.270)1.4877AcceptedIT5(0.743, 0.289, 0.311)1.3796AcceptedLO1(0.744, 0.270, 0.276)1.4677AcceptedLO2(0.756, 0.267, 0.281)1.5167AcceptedLO3(0.756, 0.267, 0.281)1.5167AcceptedLO4(0.752, 0.262, 0.269)1.5267AcceptedThreshold (Di)1.2947Figure 4
Final results of the SF-Delphi method.Additionally, the computation process of the SF-Delphi method respect with PR1 criterion is presented as follows:(28)AggPR1=α,β,γ=0.762,0.250,0.255,αPR1=∏i=1nαFSiwi=0.91/12∗0.81/12∗0.81/12∗0.71/12∗0.81/12∗0.71/12∗0.81/12∗0.91/12∗0.71/12∗0.61/12∗0.81/12∗0.71/12=0.762,βPR1=1−∏i=1n1−βFSi2wi1/2=1−1−0.121/12∗1−0.221/12∗1−0.221/12∗1−0.321/12∗1−0.221/12∗1−0.321/12∗1−0.221/12∗1−0.121/12∗1−0.321/12∗1−0.421/12∗1−0.221/12∗1−0.321/121/2=0.250γC1=∏i=1n1−βFSi2wi−∏i=1n1−βFSi2−γFSi2wi1/2=1−0.121/12∗1−0.221/12∗1−0.221/12∗1−0.321/12∗1−0.221/12∗1−0.321/12∗1−0.221/12∗1−0.121/12∗1−0.321/12∗1−0.421/12∗1−0.221/12∗1−0.321/12−1−0.12−0.121/12∗1−0.22−0.221/12∗1−0.22−0.221/12∗1−0.32−0.321/12∗1−0.22−0.221/12∗1−0.32−0.321/12∗1−0.22−0.221/12∗1−0.12−0.121/12∗1−0.32−0.321/12∗1−0.42−0.421/12∗1−0.22−0.221/12∗1−0.32−0.321/2=0.255,ScoredPR1=2αij−γij2−βij−γij2=2∗0.762−0.2552−0.2500−0.2552=1.6102.


### 4.3. Results of Subjective Weights of Criteria

Regarding SF-Delphi results, the panel of 12 experts continued to give their linguistic evaluations for 17 selected criteria as an extended round. Each selected criterion's spherical fuzzy relative preference weight was aggregated using the SWAM operator regarding equations (13)-(14), respectively. Further, defuzzied weights also were obtained using (15). Table 6 summarizes the weighted results.Table 6
Results of criteria weights and rankings.
CriteriaSF-weightsα,β,γDefuzzied weightRankingPR10.660.350.090.26211PR20.760.250.080.4411PR30.690.320.090.3128PR40.760.250.080.4333SQ10.680.340.090.27910SQ20.640.380.110.20715SQ30.690.320.100.2999SQ40.750.250.080.4234SQ60.650.370.120.22213IT10.700.300.090.3297IT20.640.370.110.21414IT40.710.300.100.3405IT50.650.370.100.22812LO10.700.300.090.3366LO20.630.390.100.20016LO30.750.250.070.4392LO40.610.400.130.15817According to the 17 selected criteria from Table6, the top five most critical criteria are PR2 > LO3 > PR4 > SQ4 > IT4. The findings suggest that when experts evaluate critical criteria in the context of package tour selection, they should emphasize transportation cost, local culture, package discount, tour operator quality, and payment method. In contrast, the list of less essential criteria also was indicated, including LO4 > LO2 > SQ2 > IT2 > SQ6. To be more explicit in Figure 5, because of variations in weights, the results also reveal that the customers and experts will pay greater attention to the highest priorities, including PR2, SQ4, IT4, LO3, among four dimensions related to price, service quality, information and technology, and location.Figure 5
Weight results in respect to 4 main dimensions.


### 4.4. Results of SF-TOPSIS

Regarding the SF-TOPSIS-SWAM operator, we aggregated a spherical fuzzy decision matrix based on decision-makers’ opinions in Table7 and presented the weighted normalized matrix in Table 8. Score function values and SF-PIS and SF-NIS are listed in Table 9. Based on Table 10, closeness ratios depict the ranking of alternatives in Table 11. We can see that the larger Table 12value of closeness ratio indicates the most preferred alternatives. In this study, the alternative X3 scores the maximum closeness ratio value of 2.829, whereas the alternative X4 scores the lowest closeness ratio value of 0.046. The ranking is obtained on the basis of surveys conducted and data analysis performed (X3 > X7 > X8 > X6 > X1 > X10 > X5 > X2 > X9 > X4).Table 7
Aggregate the judgments of DMs (SF-TOPSIS-SWAM operator).
X1X2X3X4X5X6X7X8X9X10PR1(0.58, 0.46, 0.30)(0.60, 0.42, 0.30)(0.39, 0.64, 0.26)(0.70, 0.31, 0.24)(0.70, 0.32, 0.10)(0.50, 0.53, 0.32)(0.48, 0.55, 0.33)(0.43, 0.63, 0.21)(0.80, 0.20, 0.09)(0.70, 0.30, 0.24)PR2(0.75, 0.26, 0.18)(0.77, 0.23, 0.15)(0.59, 0.46, 0.12)(0.78, 0.22, 0.19)(0.63, 0.39, 0.27)(0.80, 0.22, 0.15)(0.69, 0.31, 0.26)(0.60, 0.40, 0.37)(0.80, 0.20, 0.09)(0.80, 0.20, 0.09)PR3(0.64, 0.37, 0.35)(0.74, 0.27, 0.17)(0.61, 0.39, 0.35)(0.76, 0.24, 0.19)(0.68, 0.33, 0.29)(0.65, 0.37, 0.29)(0.530.48, 0.42)(0.75, 0.26, 0.11)(0.57, 0.43, 0.36)(0.60, 0.40, 0.36)PR4(0.57, 0.44, 0.38)(0.52, 0.49, 0.41)(0.77, 0.24, 0.06)(0.63, 0.38, 0.32)(0.63, 0.38, 0.29)(0.61, 0.40, 0.36)(0.49, 0.53, 0.25)(0.70, 0.31, 0.20)(0.79, 0.21, 0.12)(0.40, 0.60, 0.37)SQ1(0.77, 0.24, 0.15)(0.51, 0.52, 0.33)(0.63, 0.38, 0.35)(0.66, 0.36, 0.24)(0.63, 0.40, 0.23)(0.66, 0.36, 0.19)(0.58, 0.47, 0.25)(0.71, 0.30, 0.23)(0.75, 0.25, 0.21)(0.42, 0.60, 0.32)SQ2(0.63, 0.38, 0.30)(0.59, 0.43, 0.36)(0.70, 0.32, 0.23)(0.56, 0.45, 0.37)(0.74, 0.27, 0.20)(0.63, 0.39, 0.26)(0.74, 0.27, 0.18)(0.62, 0.40, 0.32)(0.63, 0.40, 0.27)(0.50, 0.52, 0.37)SQ3(0.81, 0.20, 0.07)(0.64, 0.37, 0.29)(0.65, 0.37, 0.32)(0.71, 0.30, 0.22)(0.66, 0.34, 0.27)(0.64, 0.38, 0.26)(0.66, 0.35, 0.27)(0.70, 0.32, 0.24)(0.68, 0.33, 0.25)(0.70, 0.30, 0.24)SQ4(0.69, 0.33, 0.23)(0.48, 0.53, 0.37)(0.68, 0.35, 0.27)(0.67, 0.33, 0.28)(0.65, 0.36, 0.28)(0.67, 0.35, 0.27)(0.58, 0.44, 0.22)(0.63, 0.40, 0.24)(0.60, 0.41, 0.39)(0.59, 0.41, 0.37)SQ6(0.63, 0.39, 0.27)(0.72, 0.29, 0.14)(0.65, 0.37, 0.30)(0.59, 0.43, 0.27)(0.63, 0.39, 0.30)(0.71, 0.31, 0.25)(0.55, 0.48, 0.21)(0.67, 0.35, 0.26)(0.62, 0.41, 0.27)(0.73, 0.28, 0.23)IT1(0.66, 0.37, 0.28)(0.74, 0.28, 0.12)(0.64, 0.37, 0.31)(0.67, 0.35, 0.32)(0.75, 0.26, 0.20)(0.62, 0.41, 0.27)(0.65, 0.36, 0.28)(0.61, 0.41, 0.34)(0.54, 0.50, 0.30)(0.80, 0.20, 0.11)IT2(0.70, 0.31, 0.25)(0.61, 0.40, 0.38)(0.70, 0.31, 0.23)(0.65, 0.37, 0.29)(0.67, 0.34, 0.23)(0.63, 0.40, 0.28)(0.70, 0.31, 0.27)(0.61, 0.42, 0.33)(0.67, 0.34, 0.29)(0.77, 0.23, 0.13)IT4(0.65, 0.36, 0.32)(0.81, 0.20, 0.06)(0.69, 0.33, 0.27)(0.72, 0.29, 0.29)(0.71, 0.30, 0.19)(0.69, 0.33, 0.25)(0.67, 0.34, 0.31)(0.62, 0.41, 0.13)(0.74, 0.26, 0.21)(0.66, 0.35, 0.32)IT5(0.73, 0.28, 0.22)(0.66, 0.36, 0.36)(0.66, 0.35, 0.32)(0.73, 0.28, 0.23)(0.69, 0.32, 0.26)(0.66, 0.36, 0.29)(0.67, 0.35, 0.28)(0.75, 0.27, 0.16)(0.71, 0.29, 0.23)(0.47, 0.56, 0.29)LO1(0.52, 0.51, 0.29)(0.68, 0.32, 0.28)(0.68, 0.33, 0.29)(0.75, 0.26, 0.18)(0.62, 0.41, 0.23)(0.60, 0.42, 0.24)(0.66, 0.36, 0.29)(0.75, 0.26, 0.23)(0.65, 0.35, 0.32)(0.56, 0.44, 0.42)LO2(0.71, 0.29, 0.24)(0.59, 0.42, 0.39)(0.69, 0.32, 0.28)(0.65, 0.35, 0.33)(0.63, 0.39, 0.27)(0.57, 0.44, 0.34)(0.80, 0.21, 0.17)(0.62, 0.42, 0.22)(0.76, 0.24, 0.14)(0.60, 0.40, 0.36)LO3(0.57, 0.46, 0.31)(0.47, 0.57, 0.26)(0.71, 0.29, 0.25)(0.72, 0.30, 0.11)(0.56, 0.45, 0.39)(0.62, 0.39, 0.32)(0.72, 0.28, 0.21)(0.65, 0.36, 0.32)(0.77, 0.24, 0.15)(0.70, 0.30, 0.24)LO4(0.79, 0.21, 0.12)(0.30, 0.74, 0.21)(0.81, 0.19, 0.08)(0.68, 0.33, 0.26)(0.44, 0.59, 0.32)(0.73, 0.27, 0.22)(0.58, 0.43, 0.34)(0.70, 0.30, 0.24)(0.63, 0.37, 0.33)(0.71, 0.29, 0.23)Table 8
Weighted decision matrix (SF-TOPSIS-SWAM operator).
X1X2X3X4X5X6X7X8X9X10PR1(0.38, 0.40, 0.26)(0.46, 0.47, 0.33)(0.32, 0.29, 0.53)(0.46, 0.55, 0.53)(0.69, 0.45, 0.46)(0.61, 0.62, 0.69)(0.40, 0.45, 0.29)(0.29, 0.25, 0.24)(0.12, 0.31, 0.32)(0.21, 0.12, 0.24)PR2(0.50, 0.51, 0.39)(0.52, 0.42, 0.53)(0.46, 0.40, 0.53)(0.53, 0.43, 0.41)(0.55, 0.41, 0.51)(0.41, 0.46, 0.52)(0.40, 0.40, 0.18)(0.16, 0.14, 0.20)(0.26, 0.16, 0.26)(0.36, 0.12, 0.12)PR3(0.43, 0.49, 0.41)(0.50, 0.45, 0.43)(0.35, 0.50, 0.38)(0.40, 0.49, 0.43)(0.51, 0.42, 0.47)(0.49, 0.57, 0.42)(0.54, 0.51, 0.34)(0.18, 0.33, 0.19)(0.29, 0.28, 0.40)(0.13, 0.35, 0.35)PR4(0.38, 0.35, 0.51)(0.42, 0.42, 0.41)(0.33, 0.46, 0.53)(0.27, 0.54, 0.58)(0.42, 0.50, 0.50)(0.51, 0.61, 0.45)(0.40, 0.66, 0.36)(0.39, 0.10, 0.31)(0.28, 0.34, 0.24)(0.21, 0.14, 0.35)SQ1(0.51, 0.34, 0.42)(0.44, 0.42, 0.44)(0.38, 0.47, 0.50)(0.28, 0.42, 0.60)(0.50, 0.49, 0.51)(0.49, 0.57, 0.45)(0.42, 0.67, 0.17)(0.32, 0.33, 0.24)(0.23, 0.20, 0.25)(0.23, 0.22, 0.31)SQ2(0.42, 0.39, 0.46)(0.37, 0.49, 0.42)(0.49, 0.41, 0.42)(0.33, 0.50, 0.53)(0.46, 0.55, 0.43)(0.50, 0.43, 0.51)(0.51, 0.60, 0.29)(0.35, 0.23, 0.35)(0.20, 0.26, 0.19)(0.31, 0.27, 0.35)SQ3(0.54, 0.43, 0.43)(0.47, 0.44, 0.43)(0.44, 0.46, 0.45)(0.46, 0.40, 0.49)(0.49, 0.45, 0.48)(0.500.48, 0.46)(0.47, 0.45, 0.11)(0.28, 0.31, 0.22)(0.26, 0.25, 0.26)(0.24, 0.25, 0.24)SQ4(0.46, 0.32, 0.45)(0.45, 0.43, 0.44)(0.38, 0.42, 0.40)(0.39, 0.46, 0.61)(0.48, 0.47, 0.49)(0.48, 0.54, 0.51)(0.52, 0.52, 0.23)(0.35, 0.26, 0.27)(0.28, 0.27, 0.22)(0.24, 0.37, 0.35)SQ6(0.42, 0.48, 0.43)(0.39, 0.42, 0.47)(0.37, 0.44, 0.41)(0.48, 0.50, 0.44)(0.50, 0.54, 0.50)(0.45, 0.57, 0.48)(0.52, 0.44, 0.27)(0.15, 0.30, 0.26)(0.29, 0.25, 0.21)(0.25, 0.26, 0.23)IT1(0.44, 0.49, 0.43)(0.44, 0.50, 0.41)(0.43, 0.40, 0.36)(0.53, 0.49, 0.44)(0.49, 0.48, 0.43)(0.52, 0.49, 0.52)(0.59, 0.40, 0.27)(0.14, 0.30, 0.31)(0.20, 0.26, 0.27)(0.33, 0.29, 0.13)IT2(0.47, 0.40, 0.47)(0.43, 0.45, 0.42)(0.47, 0.40, 0.45)(0.51, 0.45, 0.51)(0.46, 0.49, 0.47)(0.51, 0.45, 0.52)(0.47, 0.41, 0.25)(0.36, 0.23, 0.28)(0.23, 0.27, 0.26)(0.32, 0.28, 0.15)IT4(0.43, 0.54, 0.45)(0.48, 0.47, 0.46)(0.44, 0.41, 0.49)(0.44, 0.49, 0.40)(0.47, 0.44, 0.45)(0.47, 0.48, 0.52)(0.43, 0.48, 0.31)(0.10, 0.27, 0.28)(0.20, 0.25, 0.30)(0.15, 0.22, 0.31)IT5(0.49, 0.43, 0.44)(0.49, 0.45, 0.43)(0.45, 0.50, 0.47)(0.31, 0.44, 0.49)(0.48, 0.44, 0.46)(0.49, 0.48, 0.43)(0.44, 0.63, 0.22)(0.35, 0.31, 0.23)(0.26, 0.28, 0.27)(0.17, 0.23, 0.28)LO1(0.35, 0.45, 0.45)(0.50, 0.41, 0.40)(0.44, 0.50, 0.43)(0.37, 0.60, 0.46)(0.47, 0.43, 0.52)(0.53, 0.48, 0.42)(0.48, 0.54, 0.28)(0.27, 0.28, 0.18)(0.23, 0.24, 0.28)(0.23, 0.31, 0.40)LO2(0.47, 0.39, 0.45)(0.43, 0.42, 0.38)(0.53, 0.41, 0.51)(0.40, 0.45, 0.53)(0.46, 0.48, 0.51)(0.54, 0.40, 0.53)(0.42, 0.51, 0.24)(0.37, 0.28, 0.32)(0.27, 0.33, 0.18)(0.22, 0.16, 0.35)LO3(0.38, 0.31, 0.47)(0.48, 0.37, 0.41)(0.48, 0.43, 0.51)(0.46, 0.55, 0.64)(0.45, 0.45, 0.55)(0.51, 0.44, 0.49)(0.41, 0.45, 0.30)(0.25, 0.25, 0.14)(0.37, 0.30, 0.22)(0.31, 0.16, 0.24)LO4(0.52, 0.20, 0.54)()0.45, 0.29, 0.48)(0.38, 0.47, 0.42)(0.47, 0.40, 0.77)(0.39, 0.46, 0.65)(0.43, 0.53, 0.45)(0.49, 0.44, 0.14)(0.20, 0.11, 0.25)(0.31, 0.22, 0.32)(0.24, 0.32, 0.23)Table 9
Score function values and SF-PIS and SF-NIS (SF-TOPSIS-SWAM operator).
AlternativesX1X2X3X4X5X6X7X8X9X10maxminX∗ (best)X− (worst)PR1−0.062−0.043−0.1940.0030.004−0.089−0.096−0.2250.0910.0060.091−0.225(0.53, 0.40, 0.12)(0.29, 0.69, 0.21)PR20.0380.058−0.1090.057−0.0360.0740.001−0.0240.0910.0910.091−0.109(0.53, 0.40, 0.12)(0.39, 0.55, 0.14)PR3−0.0160.031−0.0250.046−0.006−0.022−0.0260.050−0.035−0.0250.050−0.035(0.50, 0.42, 0.13)(0.38, 0.54, 0.35)PR4−0.031−0.0320.064−0.023−0.027−0.024−0.1280.0050.082−0.0880.082−0.128(0.53, 0.40, 0.14)(0.33, 0.61, 0.24)SQ10.058−0.077−0.021−0.021−0.046−0.028−0.0830.0100.037−0.1290.058−0.129(0.51, 0.42, 0.17)(0.28, 0.67, 0.31)SQ2−0.027−0.0330.001−0.0390.030−0.0350.030−0.031−0.038−0.0620.030−0.062(0.49, 0.43, 0.20)(0.33, 0.60, 0.35)SQ30.099−0.023−0.0190.008−0.014−0.032−0.0150.000−0.0070.0060.099−0.032(0.54, 0.40, 0.11)(0.43, 0.50, 0.25)SQ4−0.003−0.067−0.011−0.009−0.020−0.015−0.078−0.044−0.021−0.026−0.003−0.078(0.46, 0.46, 0.23)(0.38, 0.54, 0.22)SQ6−0.0320.022−0.021−0.058−0.0300.007−0.103−0.015−0.0470.0210.022−0.103(0.48, 0.44, 0.15)(0.37, 0.57, 0.21)IT1−0.0200.034−0.022−0.0100.037−0.045−0.021−0.031−0.0850.0890.089−0.085(0.53, 0.40, 0.13)(0.36, 0.59, 0.29)IT20.005−0.0210.004−0.021−0.012−0.0370.004−0.034−0.0090.0610.061−0.037(0.51, 0.41, 0.15)(0.42, 0.51, 0.27)IT4−0.0180.100−0.0040.0120.010−0.004−0.011−0.0710.032−0.0130.100−0.071(0.54, 0.40, 0.10)(0.41, 0.52, 0.15)IT50.024−0.012−0.0140.021−0.003−0.018−0.0140.0390.014−0.1240.039−0.124(0.50, 0.43, 0.17)(0.31, 0.63, 0.28)LO1−0.093−0.005−0.0060.039−0.053−0.057−0.0180.033−0.015−0.0190.039−0.093(0.50, 0.43, 0.18)(0.35, 0.60, 0.28)LO20.011−0.024−0.002−0.013−0.035−0.0430.071−0.0600.054−0.0250.071−0.060(0.53, 0.40, 0.18)(0.41, 0.53, 0.22)LO3−0.057−0.1450.0110.019−0.032−0.0300.020−0.0180.0570.0060.057−0.145(0.51, 0.41, 0.16)(0.31, 0.64, 0.25)LO40.076−0.3270.103−0.006−0.1200.023−0.0410.008−0.0190.0120.103−0.327(0.54, 0.39, 0.11)(0.20, 0.77, 0.20)Table 10
Distances to PIS, NIS, and closeness ratio of each alternative based on the SWAM operator.
AlternativesD (Xi, X∗)D (Xi, X-)Closeness ratioRankX10.0630.0441.4835X20.0450.0620.6688X30.0910.0112.8291X40.0300.0730.04610X50.0490.0600.8197X60.0670.0451.6244X70.0900.0382.4592X80.0840.0601.9673X90.0460.0760.5199X100.0500.0630.8206max0.0910.076min0.0300.011Table 11
Weighted decision matrix (SF-TOPSIS-SWGM operator).
X1X2X3X4X5X6X7X8X9X10PR1(0.27, 0.33, 0.21)(0.45, 0.37, 0.27)(0.23, 0.16, 0.53)(0.46, 0.67, 0.60)(0.74, 0.47, 0.58)(0.66, 0.72, 0.79)(0.40, 0.45, 0.26)(0.31, 0.22, 0.28)(0.24, 0.32, 0.24)(0.11, 0.19, 0.27)PR2(0.45, 0.50, 0.25)(0.50, 0.32, 0.41)(0.45, 0.39, 0.53)(0.53, 0.49, 0.42)(0.71, 0.43, 0.63)(0.55, 0.47, 0.52)(0.40, 0.40, 0.23)(0.25, 0.24, 0.22)(0.17, 0.21, 0.30)(0.36, 0.19, 0.19)PR3(0.40, 0.43, 0.38)(0.50, 0.39, 0.38)(0.34, 0.45, 0.36)(0.40, 0.52, 0.51)(0.55, 0.43, 0.55)(0.56, 0.58, 0.49)(0.57, 0.51, 0.37)(0.28, 0.34, 0.23)(0.18, 0.21, 0.41)(0.29, 0.35, 0.36)PR4(0.36, 0.32, 0.46)(0.40, 0.38, 0.37)(0.26, 0.42, 0.52)(0.27, 0.56, 0.61)(0.47, 0.52, 0.54)(0.56, 0.68, 0.51)(0.41, 0.66, 0.39)(0.38, 0.27, 0.36)(0.30, 0.36, 0.28)(0.28, 0.22, 0.34)SQ1(0.44, 0.27, 0.39)(0.35, 0.32, 0.35)(0.24, 0.42, 0.48)(0.23, 0.51, 0.67)(0.53, 0.59, 0.62)(0.59, 0.71, 0.51)(0.45, 0.71, 0.24)(0.30, 0.35, 0.29)(0.28, 0.25, 0.19)(0.27, 0.28, 0.29)SQ2(0.37, 0.34, 0.41)(0.33, 0.43, 0.34)(0.45, 0.35, 0.33)(0.25, 0.56, 0.58)(0.53, 0.60, 0.50)(0.60, 0.48, 0.59)(0.61, 0.69, 0.34)(0.36, 0.29, 0.37)(0.25, 0.27, 0.28)(0.24, 0.33, 0.28)SQ3(0.50, 0.39, 0.38)(0.41, 0.42, 0.33)(0.39, 0.37, 0.41)(0.46, 0.44, 0.54)(0.55, 0.53, 0.50)(0.62, 0.54, 0.58)(0.52, 0.45, 0.29)(0.31, 0.32, 0.29)(0.34, 0.20, 0.37)(0.12, 0.28, 0.27)SQ4(0.39, 0.29, 0.34)(0.43, 0.40, 0.37)(0.31, 0.32, 0.38)(0.39, 0.54, 0.64)(0.60, 0.49, 0.53)(0.57, 0.63, 0.62)(0.54, 0.52, 0.39)(0.33, 0.17, 0.36)(0.31, 0.34, 0.30)(0.33, 0.39, 0.37)SQ6(0.35, 0.42, 0.33)(0.30, 0.36, 0.40)(0.27, 0.37, 0.31)(0.47, 0.58, 0.52)(0.62, 0.64, 0.57)(0.54, 0.67, 0.57)(0.63, 0.45, 0.34)(0.26, 0.21, 0.29)(0.28, 0.27, 0.29)(0.28, 0.22, 0.29)IT1(0.33, 0.41, 0.38)(0.38, 0.43, 0.30)(0.37, 0.33, 0.24)(0.52, 0.62, 0.54)(0.54, 0.56, 0.51)(0.65, 0.56, 0.61)(0.71, 0.41, 0.30)(0.23, 0.34, 0.30)(0.27, 0.20, 0.29)(0.30, 0.17, 0.21)IT2(0.41, 0.38, 0.39)(0.35, 0.39, 0.31)(0.41, 0.32, 0.41)(0.49, 0.52, 0.54)(0.55, 0.59, 0.54)(0.64, 0.52, 0.62)(0.52, 0.44, 0.27)(0.38, 0.29, 0.30)(0.34, 0.17, 0.20)(0.25, 0.31, 0.25)IT4(0.40, 0.46, 0.41)(0.42, 0.42, 0.37)(0.41, 0.31, 0.48)(0.42, 0.52, 0.48)(0.52, 0.51, 0.51)(0.58, 0.51, 0.64)(0.44, 0.50, 0.37)(0.30, 0.35, 0.27)(0.30, 0.29, 0.32)(0.31, 0.26, 0.34)IT5(0.44, 0.38, 0.40)(0.41, 0.44, 0.38)(0.35, 0.40, 0.47)(0.25, 0.49, 0.54)(0.52, 0.53, 0.48)(0.55, 0.59, 0.54)(0.45, 0.70, 0.29)(0.40, 0.35, 0.31)(0.34, 0.29, 0.23)(0.26, 0.27, 0.23)LO1(0.26, 0.44, 0.41)(0.47, 0.32, 0.33)(0.37, 0.47, 0.42)(0.35, 0.68, 0.48)(0.52, 0.46, 0.63)(0.60, 0.56, 0.45)(0.50, 0.56, 0.24)(0.33, 0.32, 0.28)(0.26, 0.29, 0.35)(0.24, 0.34, 0.42)LO2(0.45, 0.35, 0.44)(0.41, 0.33, 0.34)(0.44, 0.27, 0.48)(0.40, 0.48, 0.58)(0.48, 0.51, 0.61)(0.59, 0.51, 0.68)(0.45, 0.51, 0.29)(0.39, 0.34, 0.35)(0.21, 0.33, 0.27)(0.24, 0.25, 0.36)LO3(0.32, 0.23, 0.46)(0.39, 0.34, 0.38)(0.45, 0.37, 0.48)(0.46, 0.62, 0.71)(0.46, 0.56, 0.58)(0.54, 0.48, 0.57)(0.45, 0.45, 0.32)(0.22, 0.30, 0.24)(0.38, 0.32, 0.27)(0.25, 0.27, 0.27)LO4(0.52, 0.15, 0.54)(0.44, 0.23, 0.47)(0.35, 0.46, 0.41)(0.47, 0.41, 0.81)(0.40, 0.48, 0.71)(0.44, 0.58, 0.45)(0.50, 0.45, 0.21)(0.10, 0.19, 0.30)(0.25, 0.26, 0.33)(0.28, 0.34, 0.26)Table 12
Score function values and SF-PIS and SF-NIS (SF-TOPSIS-SWGM operator).
AlternativesX1X2X3X4X5X6X7X8X9X10maxminX∗ (best)X− (worst)PR1−0.168−0.085−0.266−0.008−0.101−0.118−0.225−0.4710.0730.0060.073−0.471(0.53, 0.40, 0.19)(0.16, 0.79, 0.11)PR2−0.0230.032−0.2150.035−0.187−0.075−0.004−0.0270.0730.0730.073−0.215(0.53, 0.40, 0.19)(0.25, 0.71, 0.24)PR3−0.024−0.028−0.0400.031−0.096−0.093−0.022−0.013−0.047−0.0230.031−0.096(0.50, 0.43, 0.23)(0.39, 0.55, 0.18)PR4−0.028−0.045−0.006−0.023−0.053−0.037−0.166−0.0340.052−0.0960.052−0.166(0.52, 0.41, 0.22)(0.26, 0.68, 0.28)SQ1−0.031−0.135−0.030−0.091−0.113−0.106−0.272−0.0330.011−0.1750.011−0.272(0.48, 0.45, 0.28)(0.24, 0.71, 0.19)SQ2−0.048−0.050−0.044−0.053−0.030−0.105−0.012−0.108−0.079−0.170−0.012−0.170(0.45, 0.48, 0.28)(0.25, 0.69, 0.28)SQ30.022−0.043−0.052−0.042−0.019−0.164−0.031−0.145−0.0390.0060.022−0.164(0.50, 0.44, 0.29)(0.33, 0.62, 0.20)SQ4−0.024−0.099−0.159−0.013−0.039−0.053−0.112−0.090−0.022−0.023−0.013−0.159(0.43, 0.49, 0.36)(0.34, 0.60, 0.17)SQ6−0.058−0.044−0.158−0.122−0.079−0.056−0.143−0.076−0.1640.0040.004−0.164(0.47, 0.45, 0.29)(0.31, 0.63, 0.22)IT1−0.097−0.068−0.040−0.062−0.031−0.197−0.066−0.094−0.2940.0590.059−0.294(0.52, 0.41, 0.21)(0.24, 0.71, 0.17)IT2−0.042−0.024−0.058−0.080−0.039−0.202−0.061−0.135−0.0330.0210.021−0.202(0.49, 0.44, 0.25)(0.31, 0.64, 0.17)IT4−0.021−0.006−0.026−0.037−0.027−0.079−0.029−0.1090.014−0.0200.014−0.109(0.48, 0.44, 0.26)(0.31, 0.64, 0.31)IT5−0.019−0.021−0.026−0.038−0.010−0.060−0.114−0.0610.008−0.2130.008−0.213(0.47, 0.45, 0.27)(0.25, 0.70, 0.23)LO1−0.194−0.010−0.0330.002−0.132−0.093−0.0460.010−0.019−0.0160.010−0.194(0.47, 0.45, 0.24)(0.26, 0.68, 0.24)LO2−0.010−0.031−0.011−0.020−0.141−0.063−0.026−0.1960.016−0.0230.016−0.196(0.48, 0.45, 0.25)(0.27, 0.68, 0.24)LO3−0.090−0.236−0.003−0.076−0.040−0.043−0.015−0.0850.0120.0060.012−0.236(0.48, 0.45, 0.27)(0.23, 0.71, 0.22)LO40.056−0.5020.079−0.014−0.2080.012−0.0640.004−0.0210.0090.079−0.502(0.54, 0.40, 0.19)(0.15, 0.81, 0.10)In order to better understand this paper, we used X1 as an example to introduce the calculation process of SF-TOPSIS based on SWAM operator in detail as follows:(29)AggPR1in respect toX1SWAM=α,β,γ=0.58,0.46,0.30,αPR1in respect toX1=1−∏i=1n1−α2wi1/2=1−1−0.721/12∗1−0.721/12∗1−0.321/12∗1−0.521/12∗1−0.121/12∗1−0.921/12∗1−0.421/12∗1−0.321/12∗1−0.521/12∗1−0.321/12∗1−0.721/12∗1−0.321/121/2=0.58,βPR1in respect toX1==∏i=1nβwi=0.31/12∗0.31/12∗0.71/12∗0.51/12∗0.91/12∗0.11/12∗0.61/12∗0.71/12∗0.51/12∗0.71/12∗0.31/12∗0.71/12=0.46,γPR1in respec toX1=∏i=1n1−α2wi−∏i=1n1−α2−γ2wi1/2=1−0.721/12∗1−0.721/12∗1−0.321/12∗1−0.521/12∗1−0.121/12∗1−0.921/12∗1−0.421/12∗1−0.321/12∗1−0.521/12∗1−0.421/12∗1−0.221/12∗1−0.321/12∗1−0.721/12∗1−0.321/121−0.72−0.321/12∗1−0.52−0.521/12∗1−0.92−0.121/12∗1−0.42−0.421/12∗1−0.32−0.321/12∗1−0.52−0.521/12∗1−0.32−0.321/12∗1−0.72−3.121/12∗1−0.32−0.321/12=0.30,AggPR1in respect toX1SWGM=α,β,γ=0.41,0.61,0.27,αPR1in respect toX1=∏i=1nαwi=0.71/12∗0.71/12∗0.31/12∗0.51/12∗0.11/12∗0.91/12∗0.41/12∗0.31/12∗0.51/12∗0.31/12∗0.71/12∗0.31/12=0.41,βPR1in respect toX1=1−∏i=1n1−β2wi1/2=1−1−0.321/12∗1−0.321/12∗1−0.721/12∗1−0.521/12∗1−0.921/12∗1−0.121/12∗1−0.621/12∗1−0.721/12∗1−0.521/12∗1−0.721/12∗1−0.321/12∗1−0.721/121/2=0.61,γPR1in respec toX1=∏i=1n1−β2wi−∏i=1n1−β2−γ2wi1/2=1−0.321/12∗1−0.321/12∗1−0.721/12∗1−0.521/12∗1−0.721/12∗1−0.321/12∗1−0.721/12−1−0.32−0.321/12∗1−0.32−0.321/12∗1−0.72−0.321/12∗1−0.52−0.521/12∗1−0.92−0.121/12∗1−0.12−0.121/12∗1−0.62−0.421/12∗1−0.72−0.321/12∗1−0.52−0.521/12∗1−0.72−0.321/12∗1−0.32−0.321/12∗1−0.72−0.321/12=0.27.Weighted decision matrix based on the SWAM operatorPR1in respect toX1=α,β,γ=0.38,0.55,0.29:(30)α=αPR1in respect toX1∗αweightPR1=0.58∗0.66=0.38,β=βPR1in respect toX12+βPR12−βPR1in respect toX12∗βPR12=0.462+0.352−0.462∗0.352=0.55,γ=1−βPR12∗γPR1in respect toX12+1−βPR1in respect toX12∗γPR12−γPR1in respect toX12∗γPR12=1−0.352∗0.302+1−0.462∗0.092∗0.302∗0.092=0.29.Weighted decision matrix based on the SWGM operatorPR1in respect toX1=α,β,γ=0.27,0.67,0.26:(31)α=αPR1in respect toX1∗αweightPR1=0.41∗0.66=0.27,β=βPR1in respect toX12+βweightPR12−βPR1in respect toX12∗βweightPR12=0.612+0.352−0.612∗0.352=0.67,γ=1−βweightPR12∗γPR1in respect toX12+1−βPR1in respect toX12∗γweightPR12−γPR1in respect toX12∗γweightPR12=1−0.352∗0.272+1−0.612∗0.092−0.272∗0.092=0.26.Score function values based on the SWAM operatorPR1inrespecttoX1=−0.062:(32)ScorePR1in respect toX1=α−γ2−β−γ2=0.38−0.292−0.55−0.292=−0.063.Score function values based on the SWGM operatorPR1inrespecttoX1=−0.168:(33)ScorePR1in respect toX1=α−γ2−β−γ2=0.27−0.262−0.67−0.262=−0.168.Distances to positive and negative ideal solutions based on the SWAM operator:(34)DX1,X−=12n∑i=1nαx1−αx−2+βx1−βx−2+γx1−γx−2=12∗170.38−0.292+0.55−0.692+0.29−0.212+0.50−0.392+0.43−0.552+0.18−0.142+0.43−0.382+0.49−0.542+0.34−0.352+0.33−0.262+0.54−0.612+0.36−0.242+0.51−0.282+0.42−0.672+0.17−0.312+0.42−0.332+0.50−0.602+0.29−0.352+0.54−0.432+0.40−0.502+0.11−0.252+0.46−0.382+0.46−0.542+0.23−0.222+0.37−0.312+0.50−0.572+0.27−0.212+0.44−0.362+0.49−0.592+0.27−0.292+0.47−0.422+0.45−0.512+0.25−0.272+0.43−0.412+0.49−0.522+0.37−0.152+0.49−0.312+0.44−0.632+0.22−0.282+0.35−0.352+0.60−0.602+0.28−0.282+0.41−0.272+0.45−0.532+0.24−0.222+0.38−0.312+0.55−0.642+0.30−0.252+0.52−0.202+0.40−0.772+0.14−0.202=0.044,DXi,X∗=12n∑i=1nαxi−αx∗2+βxi−βx∗2+γxi−γx∗212∗170.38−0.532+0.55−0.402+0.29−0.122+0.50−0.532+0.43−0.402+0.18−0.122+0.43−0.502+0.49−0.422+0.34−0.132+0.33−0.532+0.54−0.402+0.36−0.142+0.51−0.512+0.42−0.422+0.17−0.172+0.42−0.492+0.50−0.432+0.29−0.202+0.54−0.542+0.40−0.402+0.11−0.112+0.46−0.462+0.46−0.462+0.23−0.232+0.37−0.482+0.50−0.442+0.27−0.152+0.44−0.532+0.49−0.402+0.27−0.132+0.47−0.512+0.45−0.412+0.25−0.152+0.43−0.542+0.49−0.402+0.37−0.102+0.49−0.502+0.44−0.432+0.22−0.172+0.35−0.502+0.60−0.432+0.28−0.182+0.41−0.532+0.45−0.402+0.24−0.182+0.38−0.512+0.55−0.412+0.30−0.162+0.52−0.542+0.40−0.392+0.14−0.112=0.063,ClosenessRatioX1=ξX1=DX1,X∗DminX1,X∗−DX1,X−DmaxX1,X−=0.0630.030−0.0440.076=1.483.Further comparative analysis is conducted to assess the impact of a different aggregating method for SF-TOPSIS-SWGM. The same computations are presented in Tables11–13. From Table 14, the ranking results obtained by the proposed SWGM operator are slightly different from those using the SWAM operator. More specifically, the ordering of the alternatives has also changed a little due to the different attitudes of DMs. (X3 > X8 > X7 > X6 > X5 > X1 > X2 > X9 > X10 > X4).Table 13
Aggregate the judgments of DMs (SF-TOPSIS-SWGM operator).
X1X2X3X4X5X6X7X8X9X10PR1(0.41, 0.61, 0.27)(0.50, 0.52, 0.32)(0.31, 0.69, 0.23)(0.68, 0.34, 0.29)(0.55, 0.50, 0.25)(0.41, 0.60, 0.33)(0.35, 0.67, 0.25)(0.25, 0.76, 0.09)(0.80, 0.20, 0.18)(0.70, 0.30, 0.28)PR2(0.67, 0.37, 0.22)(0.75, 0.26, 0.25)(0.38, 0.66, 0.25)(0.75, 0.26, 0.21)(0.48, 0.56, 0.17)(0.62, 0.45, 0.20)(0.68, 0.33, 0.31)(0.59, 0.42, 0.37)(0.80, 0.20, 0.18)(0.80, 0.20, 0.18)PR3(0.60, 0.42, 0.39)(0.65, 0.39, 0.28)(0.57, 0.45, 0.36)(0.75, 0.26, 0.23)(0.58, 0.45, 0.17)(0.57, 0.46, 0.21)(0.51, 0.50, 0.44)(0.68, 0.36, 0.30)(0.54, 0.47, 0.36)(0.60, 0.40, 0.37)PR4(0.54, 0.47, 0.41)(0.48, 0.53, 0.40)(0.69, 0.34, 0.28)(0.60, 0.41, 0.38)(0.58, 0.44, 0.31)(0.56, 0.46, 0.38)(0.39, 0.63, 0.29)(0.64, 0.40, 0.28)(0.78, 0.23, 0.22)(0.40, 0.60, 0.36)SQ1(0.66, 0.39, 0.25)(0.41, 0.61, 0.32)(0.59, 0.43, 0.37)(0.52, 0.51, 0.30)(0.49, 0.55, 0.29)(0.53, 0.51, 0.26)(0.36, 0.67, 0.19)(0.64, 0.39, 0.27)(0.72, 0.30, 0.28)(0.35, 0.66, 0.30)SQ2(0.56, 0.47, 0.35)(0.52, 0.50, 0.38)(0.61, 0.42, 0.29)(0.50, 0.520.39)(0.66, 0.39, 0.26)(0.51, 0.52, 0.28)(0.68, 0.35, 0.29)(0.52, 0.50, 0.25)(0.50, 0.53, 0.34)(0.38, 0.64, 0.29)SQ3(0.75, 0.28, 0.30)(0.59, 0.57, 0.62)(0.63, 0.49, 0.58)(0.56, 0.62, 0.70)(0.43, 0.46, 0.42)(0.38, 0.55, 0.45)(0.49, 0.41, 0.30)(0.33, 0.33, 0.30)(0.35, 0.19, 0.38)(0.10, 0.29, 0.28)SQ4(0.59, 0.44, 0.41)(0.43, 0.52, 0.64)(0.60, 0.55, 0.46)(0.48, 0.57, 0.59)(0.58, 0.52, 0.37)(0.42, 0.48, 0.56)(0.55, 0.44, 0.41)(0.34, 0.16, 0.37)(0.32, 0.36, 0.31)(0.34, 0.41, 0.38)SQ6(0.53, 0.49, 0.35)(0.63, 0.49, 0.45)(0.54, 0.60, 0.41)(0.55, 0.47, 0.70)(0.41, 0.55, 0.57)(0.48, 0.44, 0.61)(0.48, 0.56, 0.31)(0.27, 0.21, 0.30)(0.29, 0.28, 0.31)(0.29, 0.22, 0.29)IT1(0.49, 0.54, 0.32)(0.61, 0.58, 0.57)(0.65, 0.45, 0.56)(0.50, 0.36, 0.79)(0.44, 0.44, 0.46)(0.39, 0.59, 0.47)(0.53, 0.66, 0.22)(0.23, 0.35, 0.31)(0.28, 0.20, 0.30)(0.31, 0.16, 0.21)IT2(0.62, 0.41, 0.28)(0.57, 0.59, 0.53)(0.59, 0.46, 0.62)(0.48, 0.62, 0.74)(0.43, 0.45, 0.50)(0.44, 0.57, 0.41)(0.55, 0.40, 0.29)(0.40, 0.30, 0.31)(0.35, 0.17, 0.19)(0.25, 0.32, 0.25)IT4(0.60, 0.41, 0.39)(0.70, 0.61, 0.63)(0.64, 0.55, 0.62)(0.46, 0.72, 0.63)(0.35, 0.41, 0.40)(0.39, 0.49, 0.40)(0.57, 0.29, 0.38)(0.31, 0.37, 0.28)(0.31, 0.30, 0.33)(0.32, 0.27, 0.35)IT5(0.66, 0.37, 0.30)(0.58, 0.60, 0.62)(0.66, 0.57, 0.53)(0.61, 0.71, 0.37)(0.44, 0.41, 0.42)(0.35, 0.45, 0.51)(0.44, 0.30, 0.64)(0.42, 0.37, 0.32)(0.35, 0.30, 0.23)(0.26, 0.27, 0.24)LO1(0.39, 0.66, 0.61)(0.70, 0.48, 0.50)(0.56, 0.71, 0.63)(0.53, 0.63, 0.35)(0.41, 0.32, 0.55)(0.52, 0.47, 0.30)(0.38, 0.47, 0.25)(0.34, 0.33, 0.29)(0.26, 0.30, 0.36)(0.24, 0.35, 0.44)LO2(0.67, 0.53, 0.66)(0.62, 0.50, 0.52)(0.67, 0.41, 0.73)(0.60, 0.34, 0.49)(0.35, 0.39, 0.53)(0.50, 0.39, 0.62)(0.29, 0.40, 0.30)(0.41, 0.35, 0.37)(0.22, 0.35, 0.28)(0.24, 0.26, 0.37)LO3(0.48, 0.35, 0.69)(0.59, 0.52, 0.58)(0.67, 0.55, 0.73)(0.70, 0.54, 0.66)(0.32, 0.46, 0.49)(0.44, 0.36, 0.47)(0.30, 0.30, 0.33)(0.23, 0.31, 0.24)(0.40, 0.34, 0.28)(0.26, 0.27, 0.28)LO4(0.78, 0.22, 0.81)(0.66, 0.35, 0.71)(0.52, 0.70, 0.62)(0.71, 0.22, 0.78)(0.19, 0.35, 0.66)(0.29, 0.50, 0.30)(0.38, 0.29, 0.21)(0.09, 0.18, 0.30)(0.26, 0.27, 0.34)(0.28, 0.35, 0.27)Table 14
Distances to PIS, NIS, and closeness ratio of each alternative based on SWGM operator.
AlternativesD (Xi, X∗)D (Xi, X-)Closeness ratioRankX10.0770.0722.3966X20.0570.0881.5057X30.1130.0374.0891X40.0260.1070.12610X50.0750.0592.4365X60.0810.0612.6364X70.0930.0752.9993X80.1040.0513.6292X90.0390.1220.5048X100.0360.1150.4699max0.1130.122min0.0260.037Additionally, the calculation process of SF-TOPSIS based on the SWGM operator in respect with X1 alternative as follows:Distances to positive and negative ideal solutions based on the SWGM operator:(35)DX1,X−=12n∑i=1nαx1−αx−2+βx1−βx−2+γx1−γx−212∗170.27−0.162+0.67−0.792+0.26−0.112+0.45−0.252+0.49−0.712+0.23−0.242+0.40−0.392+0.52−0.552+0.37−0.182+0.36−0.262+0.56−0.682+0.39−0.282+0.44−0.242+0.51−0.712+0.24−0.192+0.37−0.252+0.56−0.692+0.34−0.282+0.50−0.332+0.44−0.622+0.29−0.202+0.34−0.262+0.54−0.602+0.39−0.172+0.35−0.312+0.58−0.632+0.34−0.222+0.33−0.242+0.62−0.712+0.30−0.172+0.41−0.312+0.52−0.642+0.27−0.172+0.40−0.312+0.52−0.642+0.37−0.312+0.44−0.252+0.49−0.702+0.29−0.232+0.26−0.262+0.68−0.682+0.24−0.242+0.45−0.272+0.48−0.682+0.29−0.242+0.32−0.232+0.62−0.712+0.32−0.222+0.52−0.152+0.41−0.812+0.21−0.102=0.072,DXi,X∗=12n∑i=1nαxi−αx∗2+βxi−βx∗2+γxi−γx∗212∗170.27−0.532+0.67−0.402+0.26−0.192+0.45−0.532+0.49−0.402+0.23−0.192+0.40−0.502+0.52−0.432+0.37−0.232+0.36−0.522+0.56−0.412+0.39−0.222+0.44−0.482+0.51−0.452+0.24−0.282+0.37−0.452+0.56−0.482+0.34−0.282+0.50−0.502+0.44−0.442+0.29−0.292+0.34−0.432+0.54−0.492+0.39−0.362+0.35−0.472+0.58−0.452+0.34−0.292+0.33−0.522+0.62−0.412+0.30−0.212+0.41−0.492+0.52−0.442+0.27−0.262+0.40−0.482+0.52−0.442+0.37−0.262+0.44−0.472+0.49−0.452+0.29−0.272+0.26−0.472+0.68−0.452+0.24−0.242+0.45−0.482+0.48−0.452+0.29−0.252+0.32−0.482+0.62−0.452+0.32−0.272+0.52−0.542+0.41−0.402+0.21−0.192=0.077,ClosenessRatioX1=ξX1=DX1,X∗DminX1,X∗−DX1,X−DmaxX1,X−=0.0770.026−0.0720.122=2.369.


### 4.5. Comparative Analysis

To assess the suggested model’s validity and robustness, we conduct a comparative analysis in this part utilizing the different spherical fuzzy aggregating operators. To do this, two SWAM and SWGM operators based spherical fuzzy sets, developed by Kutlu Gündoğdu and Kahraman [34], will be utilized. The formula of these two operators is given in equation (16)-(17), respectively. It is worth noting that Figure 6 contains the final results and rating of alternatives. As seen in Tables 10 and 14, while the closeness ratios of the various spherical fuzzy aggregating operators differ, the order of options remains constant. The difference in closeness ratios is due to the spherical fuzzy aggregating operators’ nature. Decision-makers may choose amongst them depending on their type, scope, and objectives.Figure 6
Radar plot of the closeness ratio, ranking based on the SWAM, SWGM operators.To be more specific, we aim at implementing a pairwise comparison of the rankings produced by the SF-TOPSIS models. This comparison is performed by estimating Spearman’s rho correlation ccoefficient. The Spearman’s rho correlation coefficient is calculated using the following equation:(36)R=1−6∑i=1ndi2nn2−1.Topping the alternative list is X3-Vietravel, X7-Saigon Tourist, and X8- BenThanh Tourist. The values of the Spearman’s rho correlation coefﬁcient of rankings and closeness ratio are 0.891 and 0.901, which are very high, respectively. These results revealed that the proposed SF-TOPSIS with different aggregating methods (SF-TOPSIS-SWAM and SF-TOPSIS-SWGM) performs very similarly, given in Figure6 and Table 15.Table 15
Pairwise correlation of the closeness ratio, ranking based on SWAM, SWGM operators.
Ranking SWGMCloseness ratio SWGMRanking SWAM0.891Closeness ratio SWAM0.901


### 4.6. Discussions

This study offered a novel SF-Delphi and SF-TOPSIS framework as an effective tool for examining 17 package tour criteria and ten travel operator alternatives in Vietnam. To ascertain package tours’ primary characteristics and features, we invited 12 experts to conduct a survey. They determined that four categories (price, service quality, information and technology, and location) are the most crucial components that tourists consider before choosing a package tour. Decision makers play an essential role in delivering knowledge about alternatives in decision-making challenges. Due to time restrictions, a lack of information, knowledge, or data, and specialists’ imperfect knowledge of the topic, information about criteria weights is typically and primarily unknown. Many strategies have been offered in the literature to circumvent this limitation. One of the practical techniques to determining the crucial criterion is the SF-Delphi technique. When the information about the criterion is incomplete, the relative preference weights of the criteria are considered using spherical fuzzy scales and calculated using the SWAM operator.Regarding [70] the relative importance weights’ results, our study also shows that price is the most essential factor consistent with the prior studies [2, 70]. Based on the relative importance of package tour attribute, the findings showed that transportation cost and package discount criteria that customers most value besides the tour cost and insurance option with regard to price dimensions. Package tour operators should be conscious of their consumers’ price sensitivity and strive to deliver an added, nonmonetary value [2]. Accordingly, customers also concerned about tour operator quality. Tourism and service organizations have fiercely competed to attract tourists and acquire market share and profit by improving service quality and marketing guidelines to persuade tourists to return to an area [44]. A good destination package product is optimized to include a variety of tourist-friendly aspects that align with desired tourist experiences, enhance offering features, and deliver overall value [2]. The necessity of creating meaningful experiences in terms of local culture is evident such as an attribute that serves as the foundation for package tour differentiation as a good business strategy for travel operators [2]. Similarly, our study also found that operators are putting a greater emphasis on improving the tourist experience through their participation. Tourists impact tourism decision making by influencing payment methods in the context of information and technology [2, 70].The ultimate ranking of alternatives is determined in this study using the SF-TOPSIS method's closeness ratios. We used similarity between alternative and positive and negative ideal solutions regardless of distance. It is clear that our proposed method has the advantage of dealing with MCDM problems with incomplete criteria weights. It suggested that X3-Vietravel was the most prominent travel company among the ten alternative package tour operators, followed by X7-Saigon Tourist and X8- BenThanh Tourist. These are the top three companies in the Top Ten inbound and outbound travel firm rankings in actual operations. Our findings were corroborated with the recent reports from Vienamcredit–Vietnam’s leading provider of business information (https://vietnamcredit.com.vn/) [93]. These companies were evaluated and ranked based on a variety of criteria, including their financial capacity as demonstrated by their most recent financial statements; their media reputation; a survey of tourists and industry experts; and an examination of each company’s capital, market, labor, revenue growth rate, profit, and operation plan.Additionally, the results in this survey are congruent with those in the Vietnam Tourism Industry Report [94]. It is devoted to studying and evaluating Vietnam’s tourism and hospitality sectors to assist businesses and investors in making informed decisions. It provides an in-depth examination of the current situation and potential trends in Vietnam’s tourist industry. The report is organized into five significant sections: The external environment’s (economic, social, technological, and legal) impact on the tourism sector, both globally and in Vietnam. The study contains data from the GSO, the General Department of Tourists, provincial and municipal tourism departments, the Ministry of Planning and Investment and international data sources such as the World Economic Forum and the United Nations World Tourism Organization (UNWTO).


## 4.1. Description of a Case Study from Vietnam

This paper conducts a case study of Vietnam’s top ten package travel providers to validate the suggested model. Following the first evaluation, a panel of specialists (Table3) conducted surveys to determine the top 10 travel and tourist companies in 2019, including X1-Hanoitourist; X2-Pegas Tourist; X3-Vietravel; X4-TransViet Travel; X5-TST Tourist; X6-Anex Tour; X7-SaigonTourist; X8-BenThanh Tourist; X9-Datviet Tour; and X10-Exo Travel. The hierarchy of 21 critical criteria influencing a customer’s decision-making process in selecting a package tour is presented in Figure 3. The first phase deals with the critical criteria influencing customer’s decision-making process in selecting a package tour available in the literature, which is followed by the selections of the relevant dimensions (price, service quality, information and technology, and location) through the experts’ opinions with spherical fuzzy scales using SF-Delphi approach. Critical criteria relevant to package tour evaluation are investigated using the SF-TOPSIS approach to find the optimal providers in the second phase.Table 3
Expert profiles.
ExpertPositionGenderQualificationExperienceDM 1ManagerMaleMBA12DM 2ManagerFemaleMBA15DM 3ManagerMalePh.D19DM 4ManagerFemaleMBA18DM 5ManagerMaleMBA12DM 6ManagerFemalePh.D16DM 7ManagerMaleMBA13DM 8ManagerMaleMBA10DM 9Full ProfessorFemalePh.D10DM 10Distinguished ProfessorMalePost-Doctoral10DM 11Associate ProfessorMalePost-Doctoral15DM 12Associate ProfessorMalePh.D20Figure 3
Hierarchy of proposed criteria and alternatives.


## 4.2. Results of the SF-Delphi Method

The list of 21 possible indicators was compiled using secondary sources and expert consultants. The inquiry was divided into two sections and planned to be finished in 30 minutes. Section1 is devoted to the participant's demographic information. There were questions on the industry sector, a position occupied, education, and years of experience. They were then asked to express their degree of agreement with selected criteria based on their experience and expertise in the second section. To begin, invitations were sent through e-mail, and the questionnaire was sent only after approval. This study created an online questionnaire in English and Vietnamese using Google forms. The data collection process was active for three months, from May to August 2021. Twenty-six experts were contacted from various companies and academia for data collection. Out of the twenty-six experts, 12 experts agreed to be part of this study. Eight experts belong to different travel and tourism companies with more than ten years’ experience.Additionally, four experts have indulged in the research and teaching on tourism and hospitality industry for more than ten years. According to the recommendation of [92], 10 to 18 experts are to assure consensus among the participants. Table 3 summarizes the profiles of the experts. Table 4 shows how each expert described the significance of each critical criterion using spherical linguistic terms as defined in Table 2.Table 4
Spherical linguistic evaluation of criteria.
DimensionsCriteriaDM1DM2DM3DM4DM5DM6DM7DM8DM9DM10DM11DM12PricePR1AMIVHIVHIHIVHIHIVHIAMIHISMIVHIHIPR2VHIAMISMIHIAMIHISMIHISMIVHIAMIVHIPR3SMIHIAMIVHIHIVHIAMIVHIVHIHISMIHIPR4HIHIHIAMIVHIVHIHIHIVHIHIHIAMIPR5EISMIEISMISMIEIEISMISMIEIEISMIService qualitySQ1HIHIVHISMIAMISMIAMISMIHIVHIAMIVHISQ2VHISMIHIVHISMIAMIAMIHIHIVHIHIAMISQ3AMIVHISMIAMISMIVHIHIVHIHIVHIVHISMISQ4AMISMIHISMIAMIVHIAMIHIHIVHIAMIHISQ5SMIEIEISMISMIEIEISMIEIEISMIEISQ6VHIAMIVHIHIAMIHIVHIHIVHIHIHISMISQ7EISMISMIEIEISMISMIEISMISMIEISMIInformation and technologyIT1VHIHIAMIHIAMIHIVHIHISMIVHIVHISMIIT2HIVHIHIHIHIAMIHIHIVHIVHIVHIVHIIT3EISMISMISMIEIEISMISMIEIEISMISMIIT4AMIHIVHIVHIHISMIHIAMIHIHIVHIHIIT5VHIAMIEIAMIVHIAMIHIVHISMIVHIEIAMILocationLO1HIAMIHIVHIAMIVHIVHISMIHIHISMIVHILO2AMIVHIVHIHIVHIVHIEIAMIAMISMIVHIHILO3VHIHIVHIAMIVHIVHIAMIHIAMISMIVHIEILO4SMISMIAMIVHIAMIHIVHIHIVHIVHIHIVHIThen, experts’ opinions were converted into spherical fuzzy scales and aggregated using the SWGM operator in equation (13)-(14). Finally, spherical score values were defuzzified by (15). To eliminate the less important criteria, threshold (Di) = 1.2947. The detailed results of the SF-Delphi technique are shown in Table 5. Based on the comparisons between score value Sdi of each criterion and Threshold (Di), 4 criteria, including PR5; SQ5; SQ7; IT3 were rejected, and 17 critical criteria were accepted and are visualized in Figure 4.Table 5
Spherical fuzzy Delphi results.
CriteriaSFNsα,β,γScore valuesSdiDecisionsPR1(0.762, 0.250, 0.255)1.6102AcceptedPR2(0.742, 0.277, 0.285)1.4356AcceptedPR3(0.744, 0.270, 0.276)1.4677AcceptedPR4(0.755, 0.254, 0.257)1.5693AcceptedPR5(0.548, 0.454, 0.459)0.4051UnacceptedSQ1(0.742, 0.277, 0.285)1.4356AcceptedSQ2(0.751, 0.266, 0.272)1.5130AcceptedSQ3(0.743, 0.274, 0.282)1.4483AcceptedSQ4(0.759, 0.261, 0.268)1.5594AcceptedSQ5(0.539, 0.462, 0.467)0.3747UnacceptedSQ6(0.753, 0.258, 0.263)1.5477AcceptedSQ7(0.556, 0.446, 0.451)0.4377UnacceptedIT1(0.744, 0.270, 0.276)1.4677AcceptedIT2(0.756, 0.251, 0.253)1.5841AcceptedIT3(0.556, 0.446, 0.451)0.4377UnacceptedIT4(0.745, 0.266, 0.270)1.4877AcceptedIT5(0.743, 0.289, 0.311)1.3796AcceptedLO1(0.744, 0.270, 0.276)1.4677AcceptedLO2(0.756, 0.267, 0.281)1.5167AcceptedLO3(0.756, 0.267, 0.281)1.5167AcceptedLO4(0.752, 0.262, 0.269)1.5267AcceptedThreshold (Di)1.2947Figure 4
Final results of the SF-Delphi method.Additionally, the computation process of the SF-Delphi method respect with PR1 criterion is presented as follows:(28)AggPR1=α,β,γ=0.762,0.250,0.255,αPR1=∏i=1nαFSiwi=0.91/12∗0.81/12∗0.81/12∗0.71/12∗0.81/12∗0.71/12∗0.81/12∗0.91/12∗0.71/12∗0.61/12∗0.81/12∗0.71/12=0.762,βPR1=1−∏i=1n1−βFSi2wi1/2=1−1−0.121/12∗1−0.221/12∗1−0.221/12∗1−0.321/12∗1−0.221/12∗1−0.321/12∗1−0.221/12∗1−0.121/12∗1−0.321/12∗1−0.421/12∗1−0.221/12∗1−0.321/121/2=0.250γC1=∏i=1n1−βFSi2wi−∏i=1n1−βFSi2−γFSi2wi1/2=1−0.121/12∗1−0.221/12∗1−0.221/12∗1−0.321/12∗1−0.221/12∗1−0.321/12∗1−0.221/12∗1−0.121/12∗1−0.321/12∗1−0.421/12∗1−0.221/12∗1−0.321/12−1−0.12−0.121/12∗1−0.22−0.221/12∗1−0.22−0.221/12∗1−0.32−0.321/12∗1−0.22−0.221/12∗1−0.32−0.321/12∗1−0.22−0.221/12∗1−0.12−0.121/12∗1−0.32−0.321/12∗1−0.42−0.421/12∗1−0.22−0.221/12∗1−0.32−0.321/2=0.255,ScoredPR1=2αij−γij2−βij−γij2=2∗0.762−0.2552−0.2500−0.2552=1.6102.


## 4.3. Results of Subjective Weights of Criteria

Regarding SF-Delphi results, the panel of 12 experts continued to give their linguistic evaluations for 17 selected criteria as an extended round. Each selected criterion's spherical fuzzy relative preference weight was aggregated using the SWAM operator regarding equations (13)-(14), respectively. Further, defuzzied weights also were obtained using (15). Table 6 summarizes the weighted results.Table 6
Results of criteria weights and rankings.
CriteriaSF-weightsα,β,γDefuzzied weightRankingPR10.660.350.090.26211PR20.760.250.080.4411PR30.690.320.090.3128PR40.760.250.080.4333SQ10.680.340.090.27910SQ20.640.380.110.20715SQ30.690.320.100.2999SQ40.750.250.080.4234SQ60.650.370.120.22213IT10.700.300.090.3297IT20.640.370.110.21414IT40.710.300.100.3405IT50.650.370.100.22812LO10.700.300.090.3366LO20.630.390.100.20016LO30.750.250.070.4392LO40.610.400.130.15817According to the 17 selected criteria from Table6, the top five most critical criteria are PR2 > LO3 > PR4 > SQ4 > IT4. The findings suggest that when experts evaluate critical criteria in the context of package tour selection, they should emphasize transportation cost, local culture, package discount, tour operator quality, and payment method. In contrast, the list of less essential criteria also was indicated, including LO4 > LO2 > SQ2 > IT2 > SQ6. To be more explicit in Figure 5, because of variations in weights, the results also reveal that the customers and experts will pay greater attention to the highest priorities, including PR2, SQ4, IT4, LO3, among four dimensions related to price, service quality, information and technology, and location.Figure 5
Weight results in respect to 4 main dimensions.


## 4.4. Results of SF-TOPSIS

Regarding the SF-TOPSIS-SWAM operator, we aggregated a spherical fuzzy decision matrix based on decision-makers’ opinions in Table7 and presented the weighted normalized matrix in Table 8. Score function values and SF-PIS and SF-NIS are listed in Table 9. Based on Table 10, closeness ratios depict the ranking of alternatives in Table 11. We can see that the larger Table 12value of closeness ratio indicates the most preferred alternatives. In this study, the alternative X3 scores the maximum closeness ratio value of 2.829, whereas the alternative X4 scores the lowest closeness ratio value of 0.046. The ranking is obtained on the basis of surveys conducted and data analysis performed (X3 > X7 > X8 > X6 > X1 > X10 > X5 > X2 > X9 > X4).Table 7
Aggregate the judgments of DMs (SF-TOPSIS-SWAM operator).
X1X2X3X4X5X6X7X8X9X10PR1(0.58, 0.46, 0.30)(0.60, 0.42, 0.30)(0.39, 0.64, 0.26)(0.70, 0.31, 0.24)(0.70, 0.32, 0.10)(0.50, 0.53, 0.32)(0.48, 0.55, 0.33)(0.43, 0.63, 0.21)(0.80, 0.20, 0.09)(0.70, 0.30, 0.24)PR2(0.75, 0.26, 0.18)(0.77, 0.23, 0.15)(0.59, 0.46, 0.12)(0.78, 0.22, 0.19)(0.63, 0.39, 0.27)(0.80, 0.22, 0.15)(0.69, 0.31, 0.26)(0.60, 0.40, 0.37)(0.80, 0.20, 0.09)(0.80, 0.20, 0.09)PR3(0.64, 0.37, 0.35)(0.74, 0.27, 0.17)(0.61, 0.39, 0.35)(0.76, 0.24, 0.19)(0.68, 0.33, 0.29)(0.65, 0.37, 0.29)(0.530.48, 0.42)(0.75, 0.26, 0.11)(0.57, 0.43, 0.36)(0.60, 0.40, 0.36)PR4(0.57, 0.44, 0.38)(0.52, 0.49, 0.41)(0.77, 0.24, 0.06)(0.63, 0.38, 0.32)(0.63, 0.38, 0.29)(0.61, 0.40, 0.36)(0.49, 0.53, 0.25)(0.70, 0.31, 0.20)(0.79, 0.21, 0.12)(0.40, 0.60, 0.37)SQ1(0.77, 0.24, 0.15)(0.51, 0.52, 0.33)(0.63, 0.38, 0.35)(0.66, 0.36, 0.24)(0.63, 0.40, 0.23)(0.66, 0.36, 0.19)(0.58, 0.47, 0.25)(0.71, 0.30, 0.23)(0.75, 0.25, 0.21)(0.42, 0.60, 0.32)SQ2(0.63, 0.38, 0.30)(0.59, 0.43, 0.36)(0.70, 0.32, 0.23)(0.56, 0.45, 0.37)(0.74, 0.27, 0.20)(0.63, 0.39, 0.26)(0.74, 0.27, 0.18)(0.62, 0.40, 0.32)(0.63, 0.40, 0.27)(0.50, 0.52, 0.37)SQ3(0.81, 0.20, 0.07)(0.64, 0.37, 0.29)(0.65, 0.37, 0.32)(0.71, 0.30, 0.22)(0.66, 0.34, 0.27)(0.64, 0.38, 0.26)(0.66, 0.35, 0.27)(0.70, 0.32, 0.24)(0.68, 0.33, 0.25)(0.70, 0.30, 0.24)SQ4(0.69, 0.33, 0.23)(0.48, 0.53, 0.37)(0.68, 0.35, 0.27)(0.67, 0.33, 0.28)(0.65, 0.36, 0.28)(0.67, 0.35, 0.27)(0.58, 0.44, 0.22)(0.63, 0.40, 0.24)(0.60, 0.41, 0.39)(0.59, 0.41, 0.37)SQ6(0.63, 0.39, 0.27)(0.72, 0.29, 0.14)(0.65, 0.37, 0.30)(0.59, 0.43, 0.27)(0.63, 0.39, 0.30)(0.71, 0.31, 0.25)(0.55, 0.48, 0.21)(0.67, 0.35, 0.26)(0.62, 0.41, 0.27)(0.73, 0.28, 0.23)IT1(0.66, 0.37, 0.28)(0.74, 0.28, 0.12)(0.64, 0.37, 0.31)(0.67, 0.35, 0.32)(0.75, 0.26, 0.20)(0.62, 0.41, 0.27)(0.65, 0.36, 0.28)(0.61, 0.41, 0.34)(0.54, 0.50, 0.30)(0.80, 0.20, 0.11)IT2(0.70, 0.31, 0.25)(0.61, 0.40, 0.38)(0.70, 0.31, 0.23)(0.65, 0.37, 0.29)(0.67, 0.34, 0.23)(0.63, 0.40, 0.28)(0.70, 0.31, 0.27)(0.61, 0.42, 0.33)(0.67, 0.34, 0.29)(0.77, 0.23, 0.13)IT4(0.65, 0.36, 0.32)(0.81, 0.20, 0.06)(0.69, 0.33, 0.27)(0.72, 0.29, 0.29)(0.71, 0.30, 0.19)(0.69, 0.33, 0.25)(0.67, 0.34, 0.31)(0.62, 0.41, 0.13)(0.74, 0.26, 0.21)(0.66, 0.35, 0.32)IT5(0.73, 0.28, 0.22)(0.66, 0.36, 0.36)(0.66, 0.35, 0.32)(0.73, 0.28, 0.23)(0.69, 0.32, 0.26)(0.66, 0.36, 0.29)(0.67, 0.35, 0.28)(0.75, 0.27, 0.16)(0.71, 0.29, 0.23)(0.47, 0.56, 0.29)LO1(0.52, 0.51, 0.29)(0.68, 0.32, 0.28)(0.68, 0.33, 0.29)(0.75, 0.26, 0.18)(0.62, 0.41, 0.23)(0.60, 0.42, 0.24)(0.66, 0.36, 0.29)(0.75, 0.26, 0.23)(0.65, 0.35, 0.32)(0.56, 0.44, 0.42)LO2(0.71, 0.29, 0.24)(0.59, 0.42, 0.39)(0.69, 0.32, 0.28)(0.65, 0.35, 0.33)(0.63, 0.39, 0.27)(0.57, 0.44, 0.34)(0.80, 0.21, 0.17)(0.62, 0.42, 0.22)(0.76, 0.24, 0.14)(0.60, 0.40, 0.36)LO3(0.57, 0.46, 0.31)(0.47, 0.57, 0.26)(0.71, 0.29, 0.25)(0.72, 0.30, 0.11)(0.56, 0.45, 0.39)(0.62, 0.39, 0.32)(0.72, 0.28, 0.21)(0.65, 0.36, 0.32)(0.77, 0.24, 0.15)(0.70, 0.30, 0.24)LO4(0.79, 0.21, 0.12)(0.30, 0.74, 0.21)(0.81, 0.19, 0.08)(0.68, 0.33, 0.26)(0.44, 0.59, 0.32)(0.73, 0.27, 0.22)(0.58, 0.43, 0.34)(0.70, 0.30, 0.24)(0.63, 0.37, 0.33)(0.71, 0.29, 0.23)Table 8
Weighted decision matrix (SF-TOPSIS-SWAM operator).
X1X2X3X4X5X6X7X8X9X10PR1(0.38, 0.40, 0.26)(0.46, 0.47, 0.33)(0.32, 0.29, 0.53)(0.46, 0.55, 0.53)(0.69, 0.45, 0.46)(0.61, 0.62, 0.69)(0.40, 0.45, 0.29)(0.29, 0.25, 0.24)(0.12, 0.31, 0.32)(0.21, 0.12, 0.24)PR2(0.50, 0.51, 0.39)(0.52, 0.42, 0.53)(0.46, 0.40, 0.53)(0.53, 0.43, 0.41)(0.55, 0.41, 0.51)(0.41, 0.46, 0.52)(0.40, 0.40, 0.18)(0.16, 0.14, 0.20)(0.26, 0.16, 0.26)(0.36, 0.12, 0.12)PR3(0.43, 0.49, 0.41)(0.50, 0.45, 0.43)(0.35, 0.50, 0.38)(0.40, 0.49, 0.43)(0.51, 0.42, 0.47)(0.49, 0.57, 0.42)(0.54, 0.51, 0.34)(0.18, 0.33, 0.19)(0.29, 0.28, 0.40)(0.13, 0.35, 0.35)PR4(0.38, 0.35, 0.51)(0.42, 0.42, 0.41)(0.33, 0.46, 0.53)(0.27, 0.54, 0.58)(0.42, 0.50, 0.50)(0.51, 0.61, 0.45)(0.40, 0.66, 0.36)(0.39, 0.10, 0.31)(0.28, 0.34, 0.24)(0.21, 0.14, 0.35)SQ1(0.51, 0.34, 0.42)(0.44, 0.42, 0.44)(0.38, 0.47, 0.50)(0.28, 0.42, 0.60)(0.50, 0.49, 0.51)(0.49, 0.57, 0.45)(0.42, 0.67, 0.17)(0.32, 0.33, 0.24)(0.23, 0.20, 0.25)(0.23, 0.22, 0.31)SQ2(0.42, 0.39, 0.46)(0.37, 0.49, 0.42)(0.49, 0.41, 0.42)(0.33, 0.50, 0.53)(0.46, 0.55, 0.43)(0.50, 0.43, 0.51)(0.51, 0.60, 0.29)(0.35, 0.23, 0.35)(0.20, 0.26, 0.19)(0.31, 0.27, 0.35)SQ3(0.54, 0.43, 0.43)(0.47, 0.44, 0.43)(0.44, 0.46, 0.45)(0.46, 0.40, 0.49)(0.49, 0.45, 0.48)(0.500.48, 0.46)(0.47, 0.45, 0.11)(0.28, 0.31, 0.22)(0.26, 0.25, 0.26)(0.24, 0.25, 0.24)SQ4(0.46, 0.32, 0.45)(0.45, 0.43, 0.44)(0.38, 0.42, 0.40)(0.39, 0.46, 0.61)(0.48, 0.47, 0.49)(0.48, 0.54, 0.51)(0.52, 0.52, 0.23)(0.35, 0.26, 0.27)(0.28, 0.27, 0.22)(0.24, 0.37, 0.35)SQ6(0.42, 0.48, 0.43)(0.39, 0.42, 0.47)(0.37, 0.44, 0.41)(0.48, 0.50, 0.44)(0.50, 0.54, 0.50)(0.45, 0.57, 0.48)(0.52, 0.44, 0.27)(0.15, 0.30, 0.26)(0.29, 0.25, 0.21)(0.25, 0.26, 0.23)IT1(0.44, 0.49, 0.43)(0.44, 0.50, 0.41)(0.43, 0.40, 0.36)(0.53, 0.49, 0.44)(0.49, 0.48, 0.43)(0.52, 0.49, 0.52)(0.59, 0.40, 0.27)(0.14, 0.30, 0.31)(0.20, 0.26, 0.27)(0.33, 0.29, 0.13)IT2(0.47, 0.40, 0.47)(0.43, 0.45, 0.42)(0.47, 0.40, 0.45)(0.51, 0.45, 0.51)(0.46, 0.49, 0.47)(0.51, 0.45, 0.52)(0.47, 0.41, 0.25)(0.36, 0.23, 0.28)(0.23, 0.27, 0.26)(0.32, 0.28, 0.15)IT4(0.43, 0.54, 0.45)(0.48, 0.47, 0.46)(0.44, 0.41, 0.49)(0.44, 0.49, 0.40)(0.47, 0.44, 0.45)(0.47, 0.48, 0.52)(0.43, 0.48, 0.31)(0.10, 0.27, 0.28)(0.20, 0.25, 0.30)(0.15, 0.22, 0.31)IT5(0.49, 0.43, 0.44)(0.49, 0.45, 0.43)(0.45, 0.50, 0.47)(0.31, 0.44, 0.49)(0.48, 0.44, 0.46)(0.49, 0.48, 0.43)(0.44, 0.63, 0.22)(0.35, 0.31, 0.23)(0.26, 0.28, 0.27)(0.17, 0.23, 0.28)LO1(0.35, 0.45, 0.45)(0.50, 0.41, 0.40)(0.44, 0.50, 0.43)(0.37, 0.60, 0.46)(0.47, 0.43, 0.52)(0.53, 0.48, 0.42)(0.48, 0.54, 0.28)(0.27, 0.28, 0.18)(0.23, 0.24, 0.28)(0.23, 0.31, 0.40)LO2(0.47, 0.39, 0.45)(0.43, 0.42, 0.38)(0.53, 0.41, 0.51)(0.40, 0.45, 0.53)(0.46, 0.48, 0.51)(0.54, 0.40, 0.53)(0.42, 0.51, 0.24)(0.37, 0.28, 0.32)(0.27, 0.33, 0.18)(0.22, 0.16, 0.35)LO3(0.38, 0.31, 0.47)(0.48, 0.37, 0.41)(0.48, 0.43, 0.51)(0.46, 0.55, 0.64)(0.45, 0.45, 0.55)(0.51, 0.44, 0.49)(0.41, 0.45, 0.30)(0.25, 0.25, 0.14)(0.37, 0.30, 0.22)(0.31, 0.16, 0.24)LO4(0.52, 0.20, 0.54)()0.45, 0.29, 0.48)(0.38, 0.47, 0.42)(0.47, 0.40, 0.77)(0.39, 0.46, 0.65)(0.43, 0.53, 0.45)(0.49, 0.44, 0.14)(0.20, 0.11, 0.25)(0.31, 0.22, 0.32)(0.24, 0.32, 0.23)Table 9
Score function values and SF-PIS and SF-NIS (SF-TOPSIS-SWAM operator).
AlternativesX1X2X3X4X5X6X7X8X9X10maxminX∗ (best)X− (worst)PR1−0.062−0.043−0.1940.0030.004−0.089−0.096−0.2250.0910.0060.091−0.225(0.53, 0.40, 0.12)(0.29, 0.69, 0.21)PR20.0380.058−0.1090.057−0.0360.0740.001−0.0240.0910.0910.091−0.109(0.53, 0.40, 0.12)(0.39, 0.55, 0.14)PR3−0.0160.031−0.0250.046−0.006−0.022−0.0260.050−0.035−0.0250.050−0.035(0.50, 0.42, 0.13)(0.38, 0.54, 0.35)PR4−0.031−0.0320.064−0.023−0.027−0.024−0.1280.0050.082−0.0880.082−0.128(0.53, 0.40, 0.14)(0.33, 0.61, 0.24)SQ10.058−0.077−0.021−0.021−0.046−0.028−0.0830.0100.037−0.1290.058−0.129(0.51, 0.42, 0.17)(0.28, 0.67, 0.31)SQ2−0.027−0.0330.001−0.0390.030−0.0350.030−0.031−0.038−0.0620.030−0.062(0.49, 0.43, 0.20)(0.33, 0.60, 0.35)SQ30.099−0.023−0.0190.008−0.014−0.032−0.0150.000−0.0070.0060.099−0.032(0.54, 0.40, 0.11)(0.43, 0.50, 0.25)SQ4−0.003−0.067−0.011−0.009−0.020−0.015−0.078−0.044−0.021−0.026−0.003−0.078(0.46, 0.46, 0.23)(0.38, 0.54, 0.22)SQ6−0.0320.022−0.021−0.058−0.0300.007−0.103−0.015−0.0470.0210.022−0.103(0.48, 0.44, 0.15)(0.37, 0.57, 0.21)IT1−0.0200.034−0.022−0.0100.037−0.045−0.021−0.031−0.0850.0890.089−0.085(0.53, 0.40, 0.13)(0.36, 0.59, 0.29)IT20.005−0.0210.004−0.021−0.012−0.0370.004−0.034−0.0090.0610.061−0.037(0.51, 0.41, 0.15)(0.42, 0.51, 0.27)IT4−0.0180.100−0.0040.0120.010−0.004−0.011−0.0710.032−0.0130.100−0.071(0.54, 0.40, 0.10)(0.41, 0.52, 0.15)IT50.024−0.012−0.0140.021−0.003−0.018−0.0140.0390.014−0.1240.039−0.124(0.50, 0.43, 0.17)(0.31, 0.63, 0.28)LO1−0.093−0.005−0.0060.039−0.053−0.057−0.0180.033−0.015−0.0190.039−0.093(0.50, 0.43, 0.18)(0.35, 0.60, 0.28)LO20.011−0.024−0.002−0.013−0.035−0.0430.071−0.0600.054−0.0250.071−0.060(0.53, 0.40, 0.18)(0.41, 0.53, 0.22)LO3−0.057−0.1450.0110.019−0.032−0.0300.020−0.0180.0570.0060.057−0.145(0.51, 0.41, 0.16)(0.31, 0.64, 0.25)LO40.076−0.3270.103−0.006−0.1200.023−0.0410.008−0.0190.0120.103−0.327(0.54, 0.39, 0.11)(0.20, 0.77, 0.20)Table 10
Distances to PIS, NIS, and closeness ratio of each alternative based on the SWAM operator.
AlternativesD (Xi, X∗)D (Xi, X-)Closeness ratioRankX10.0630.0441.4835X20.0450.0620.6688X30.0910.0112.8291X40.0300.0730.04610X50.0490.0600.8197X60.0670.0451.6244X70.0900.0382.4592X80.0840.0601.9673X90.0460.0760.5199X100.0500.0630.8206max0.0910.076min0.0300.011Table 11
Weighted decision matrix (SF-TOPSIS-SWGM operator).
X1X2X3X4X5X6X7X8X9X10PR1(0.27, 0.33, 0.21)(0.45, 0.37, 0.27)(0.23, 0.16, 0.53)(0.46, 0.67, 0.60)(0.74, 0.47, 0.58)(0.66, 0.72, 0.79)(0.40, 0.45, 0.26)(0.31, 0.22, 0.28)(0.24, 0.32, 0.24)(0.11, 0.19, 0.27)PR2(0.45, 0.50, 0.25)(0.50, 0.32, 0.41)(0.45, 0.39, 0.53)(0.53, 0.49, 0.42)(0.71, 0.43, 0.63)(0.55, 0.47, 0.52)(0.40, 0.40, 0.23)(0.25, 0.24, 0.22)(0.17, 0.21, 0.30)(0.36, 0.19, 0.19)PR3(0.40, 0.43, 0.38)(0.50, 0.39, 0.38)(0.34, 0.45, 0.36)(0.40, 0.52, 0.51)(0.55, 0.43, 0.55)(0.56, 0.58, 0.49)(0.57, 0.51, 0.37)(0.28, 0.34, 0.23)(0.18, 0.21, 0.41)(0.29, 0.35, 0.36)PR4(0.36, 0.32, 0.46)(0.40, 0.38, 0.37)(0.26, 0.42, 0.52)(0.27, 0.56, 0.61)(0.47, 0.52, 0.54)(0.56, 0.68, 0.51)(0.41, 0.66, 0.39)(0.38, 0.27, 0.36)(0.30, 0.36, 0.28)(0.28, 0.22, 0.34)SQ1(0.44, 0.27, 0.39)(0.35, 0.32, 0.35)(0.24, 0.42, 0.48)(0.23, 0.51, 0.67)(0.53, 0.59, 0.62)(0.59, 0.71, 0.51)(0.45, 0.71, 0.24)(0.30, 0.35, 0.29)(0.28, 0.25, 0.19)(0.27, 0.28, 0.29)SQ2(0.37, 0.34, 0.41)(0.33, 0.43, 0.34)(0.45, 0.35, 0.33)(0.25, 0.56, 0.58)(0.53, 0.60, 0.50)(0.60, 0.48, 0.59)(0.61, 0.69, 0.34)(0.36, 0.29, 0.37)(0.25, 0.27, 0.28)(0.24, 0.33, 0.28)SQ3(0.50, 0.39, 0.38)(0.41, 0.42, 0.33)(0.39, 0.37, 0.41)(0.46, 0.44, 0.54)(0.55, 0.53, 0.50)(0.62, 0.54, 0.58)(0.52, 0.45, 0.29)(0.31, 0.32, 0.29)(0.34, 0.20, 0.37)(0.12, 0.28, 0.27)SQ4(0.39, 0.29, 0.34)(0.43, 0.40, 0.37)(0.31, 0.32, 0.38)(0.39, 0.54, 0.64)(0.60, 0.49, 0.53)(0.57, 0.63, 0.62)(0.54, 0.52, 0.39)(0.33, 0.17, 0.36)(0.31, 0.34, 0.30)(0.33, 0.39, 0.37)SQ6(0.35, 0.42, 0.33)(0.30, 0.36, 0.40)(0.27, 0.37, 0.31)(0.47, 0.58, 0.52)(0.62, 0.64, 0.57)(0.54, 0.67, 0.57)(0.63, 0.45, 0.34)(0.26, 0.21, 0.29)(0.28, 0.27, 0.29)(0.28, 0.22, 0.29)IT1(0.33, 0.41, 0.38)(0.38, 0.43, 0.30)(0.37, 0.33, 0.24)(0.52, 0.62, 0.54)(0.54, 0.56, 0.51)(0.65, 0.56, 0.61)(0.71, 0.41, 0.30)(0.23, 0.34, 0.30)(0.27, 0.20, 0.29)(0.30, 0.17, 0.21)IT2(0.41, 0.38, 0.39)(0.35, 0.39, 0.31)(0.41, 0.32, 0.41)(0.49, 0.52, 0.54)(0.55, 0.59, 0.54)(0.64, 0.52, 0.62)(0.52, 0.44, 0.27)(0.38, 0.29, 0.30)(0.34, 0.17, 0.20)(0.25, 0.31, 0.25)IT4(0.40, 0.46, 0.41)(0.42, 0.42, 0.37)(0.41, 0.31, 0.48)(0.42, 0.52, 0.48)(0.52, 0.51, 0.51)(0.58, 0.51, 0.64)(0.44, 0.50, 0.37)(0.30, 0.35, 0.27)(0.30, 0.29, 0.32)(0.31, 0.26, 0.34)IT5(0.44, 0.38, 0.40)(0.41, 0.44, 0.38)(0.35, 0.40, 0.47)(0.25, 0.49, 0.54)(0.52, 0.53, 0.48)(0.55, 0.59, 0.54)(0.45, 0.70, 0.29)(0.40, 0.35, 0.31)(0.34, 0.29, 0.23)(0.26, 0.27, 0.23)LO1(0.26, 0.44, 0.41)(0.47, 0.32, 0.33)(0.37, 0.47, 0.42)(0.35, 0.68, 0.48)(0.52, 0.46, 0.63)(0.60, 0.56, 0.45)(0.50, 0.56, 0.24)(0.33, 0.32, 0.28)(0.26, 0.29, 0.35)(0.24, 0.34, 0.42)LO2(0.45, 0.35, 0.44)(0.41, 0.33, 0.34)(0.44, 0.27, 0.48)(0.40, 0.48, 0.58)(0.48, 0.51, 0.61)(0.59, 0.51, 0.68)(0.45, 0.51, 0.29)(0.39, 0.34, 0.35)(0.21, 0.33, 0.27)(0.24, 0.25, 0.36)LO3(0.32, 0.23, 0.46)(0.39, 0.34, 0.38)(0.45, 0.37, 0.48)(0.46, 0.62, 0.71)(0.46, 0.56, 0.58)(0.54, 0.48, 0.57)(0.45, 0.45, 0.32)(0.22, 0.30, 0.24)(0.38, 0.32, 0.27)(0.25, 0.27, 0.27)LO4(0.52, 0.15, 0.54)(0.44, 0.23, 0.47)(0.35, 0.46, 0.41)(0.47, 0.41, 0.81)(0.40, 0.48, 0.71)(0.44, 0.58, 0.45)(0.50, 0.45, 0.21)(0.10, 0.19, 0.30)(0.25, 0.26, 0.33)(0.28, 0.34, 0.26)Table 12
Score function values and SF-PIS and SF-NIS (SF-TOPSIS-SWGM operator).
AlternativesX1X2X3X4X5X6X7X8X9X10maxminX∗ (best)X− (worst)PR1−0.168−0.085−0.266−0.008−0.101−0.118−0.225−0.4710.0730.0060.073−0.471(0.53, 0.40, 0.19)(0.16, 0.79, 0.11)PR2−0.0230.032−0.2150.035−0.187−0.075−0.004−0.0270.0730.0730.073−0.215(0.53, 0.40, 0.19)(0.25, 0.71, 0.24)PR3−0.024−0.028−0.0400.031−0.096−0.093−0.022−0.013−0.047−0.0230.031−0.096(0.50, 0.43, 0.23)(0.39, 0.55, 0.18)PR4−0.028−0.045−0.006−0.023−0.053−0.037−0.166−0.0340.052−0.0960.052−0.166(0.52, 0.41, 0.22)(0.26, 0.68, 0.28)SQ1−0.031−0.135−0.030−0.091−0.113−0.106−0.272−0.0330.011−0.1750.011−0.272(0.48, 0.45, 0.28)(0.24, 0.71, 0.19)SQ2−0.048−0.050−0.044−0.053−0.030−0.105−0.012−0.108−0.079−0.170−0.012−0.170(0.45, 0.48, 0.28)(0.25, 0.69, 0.28)SQ30.022−0.043−0.052−0.042−0.019−0.164−0.031−0.145−0.0390.0060.022−0.164(0.50, 0.44, 0.29)(0.33, 0.62, 0.20)SQ4−0.024−0.099−0.159−0.013−0.039−0.053−0.112−0.090−0.022−0.023−0.013−0.159(0.43, 0.49, 0.36)(0.34, 0.60, 0.17)SQ6−0.058−0.044−0.158−0.122−0.079−0.056−0.143−0.076−0.1640.0040.004−0.164(0.47, 0.45, 0.29)(0.31, 0.63, 0.22)IT1−0.097−0.068−0.040−0.062−0.031−0.197−0.066−0.094−0.2940.0590.059−0.294(0.52, 0.41, 0.21)(0.24, 0.71, 0.17)IT2−0.042−0.024−0.058−0.080−0.039−0.202−0.061−0.135−0.0330.0210.021−0.202(0.49, 0.44, 0.25)(0.31, 0.64, 0.17)IT4−0.021−0.006−0.026−0.037−0.027−0.079−0.029−0.1090.014−0.0200.014−0.109(0.48, 0.44, 0.26)(0.31, 0.64, 0.31)IT5−0.019−0.021−0.026−0.038−0.010−0.060−0.114−0.0610.008−0.2130.008−0.213(0.47, 0.45, 0.27)(0.25, 0.70, 0.23)LO1−0.194−0.010−0.0330.002−0.132−0.093−0.0460.010−0.019−0.0160.010−0.194(0.47, 0.45, 0.24)(0.26, 0.68, 0.24)LO2−0.010−0.031−0.011−0.020−0.141−0.063−0.026−0.1960.016−0.0230.016−0.196(0.48, 0.45, 0.25)(0.27, 0.68, 0.24)LO3−0.090−0.236−0.003−0.076−0.040−0.043−0.015−0.0850.0120.0060.012−0.236(0.48, 0.45, 0.27)(0.23, 0.71, 0.22)LO40.056−0.5020.079−0.014−0.2080.012−0.0640.004−0.0210.0090.079−0.502(0.54, 0.40, 0.19)(0.15, 0.81, 0.10)In order to better understand this paper, we used X1 as an example to introduce the calculation process of SF-TOPSIS based on SWAM operator in detail as follows:(29)AggPR1in respect toX1SWAM=α,β,γ=0.58,0.46,0.30,αPR1in respect toX1=1−∏i=1n1−α2wi1/2=1−1−0.721/12∗1−0.721/12∗1−0.321/12∗1−0.521/12∗1−0.121/12∗1−0.921/12∗1−0.421/12∗1−0.321/12∗1−0.521/12∗1−0.321/12∗1−0.721/12∗1−0.321/121/2=0.58,βPR1in respect toX1==∏i=1nβwi=0.31/12∗0.31/12∗0.71/12∗0.51/12∗0.91/12∗0.11/12∗0.61/12∗0.71/12∗0.51/12∗0.71/12∗0.31/12∗0.71/12=0.46,γPR1in respec toX1=∏i=1n1−α2wi−∏i=1n1−α2−γ2wi1/2=1−0.721/12∗1−0.721/12∗1−0.321/12∗1−0.521/12∗1−0.121/12∗1−0.921/12∗1−0.421/12∗1−0.321/12∗1−0.521/12∗1−0.421/12∗1−0.221/12∗1−0.321/12∗1−0.721/12∗1−0.321/121−0.72−0.321/12∗1−0.52−0.521/12∗1−0.92−0.121/12∗1−0.42−0.421/12∗1−0.32−0.321/12∗1−0.52−0.521/12∗1−0.32−0.321/12∗1−0.72−3.121/12∗1−0.32−0.321/12=0.30,AggPR1in respect toX1SWGM=α,β,γ=0.41,0.61,0.27,αPR1in respect toX1=∏i=1nαwi=0.71/12∗0.71/12∗0.31/12∗0.51/12∗0.11/12∗0.91/12∗0.41/12∗0.31/12∗0.51/12∗0.31/12∗0.71/12∗0.31/12=0.41,βPR1in respect toX1=1−∏i=1n1−β2wi1/2=1−1−0.321/12∗1−0.321/12∗1−0.721/12∗1−0.521/12∗1−0.921/12∗1−0.121/12∗1−0.621/12∗1−0.721/12∗1−0.521/12∗1−0.721/12∗1−0.321/12∗1−0.721/121/2=0.61,γPR1in respec toX1=∏i=1n1−β2wi−∏i=1n1−β2−γ2wi1/2=1−0.321/12∗1−0.321/12∗1−0.721/12∗1−0.521/12∗1−0.721/12∗1−0.321/12∗1−0.721/12−1−0.32−0.321/12∗1−0.32−0.321/12∗1−0.72−0.321/12∗1−0.52−0.521/12∗1−0.92−0.121/12∗1−0.12−0.121/12∗1−0.62−0.421/12∗1−0.72−0.321/12∗1−0.52−0.521/12∗1−0.72−0.321/12∗1−0.32−0.321/12∗1−0.72−0.321/12=0.27.Weighted decision matrix based on the SWAM operatorPR1in respect toX1=α,β,γ=0.38,0.55,0.29:(30)α=αPR1in respect toX1∗αweightPR1=0.58∗0.66=0.38,β=βPR1in respect toX12+βPR12−βPR1in respect toX12∗βPR12=0.462+0.352−0.462∗0.352=0.55,γ=1−βPR12∗γPR1in respect toX12+1−βPR1in respect toX12∗γPR12−γPR1in respect toX12∗γPR12=1−0.352∗0.302+1−0.462∗0.092∗0.302∗0.092=0.29.Weighted decision matrix based on the SWGM operatorPR1in respect toX1=α,β,γ=0.27,0.67,0.26:(31)α=αPR1in respect toX1∗αweightPR1=0.41∗0.66=0.27,β=βPR1in respect toX12+βweightPR12−βPR1in respect toX12∗βweightPR12=0.612+0.352−0.612∗0.352=0.67,γ=1−βweightPR12∗γPR1in respect toX12+1−βPR1in respect toX12∗γweightPR12−γPR1in respect toX12∗γweightPR12=1−0.352∗0.272+1−0.612∗0.092−0.272∗0.092=0.26.Score function values based on the SWAM operatorPR1inrespecttoX1=−0.062:(32)ScorePR1in respect toX1=α−γ2−β−γ2=0.38−0.292−0.55−0.292=−0.063.Score function values based on the SWGM operatorPR1inrespecttoX1=−0.168:(33)ScorePR1in respect toX1=α−γ2−β−γ2=0.27−0.262−0.67−0.262=−0.168.Distances to positive and negative ideal solutions based on the SWAM operator:(34)DX1,X−=12n∑i=1nαx1−αx−2+βx1−βx−2+γx1−γx−2=12∗170.38−0.292+0.55−0.692+0.29−0.212+0.50−0.392+0.43−0.552+0.18−0.142+0.43−0.382+0.49−0.542+0.34−0.352+0.33−0.262+0.54−0.612+0.36−0.242+0.51−0.282+0.42−0.672+0.17−0.312+0.42−0.332+0.50−0.602+0.29−0.352+0.54−0.432+0.40−0.502+0.11−0.252+0.46−0.382+0.46−0.542+0.23−0.222+0.37−0.312+0.50−0.572+0.27−0.212+0.44−0.362+0.49−0.592+0.27−0.292+0.47−0.422+0.45−0.512+0.25−0.272+0.43−0.412+0.49−0.522+0.37−0.152+0.49−0.312+0.44−0.632+0.22−0.282+0.35−0.352+0.60−0.602+0.28−0.282+0.41−0.272+0.45−0.532+0.24−0.222+0.38−0.312+0.55−0.642+0.30−0.252+0.52−0.202+0.40−0.772+0.14−0.202=0.044,DXi,X∗=12n∑i=1nαxi−αx∗2+βxi−βx∗2+γxi−γx∗212∗170.38−0.532+0.55−0.402+0.29−0.122+0.50−0.532+0.43−0.402+0.18−0.122+0.43−0.502+0.49−0.422+0.34−0.132+0.33−0.532+0.54−0.402+0.36−0.142+0.51−0.512+0.42−0.422+0.17−0.172+0.42−0.492+0.50−0.432+0.29−0.202+0.54−0.542+0.40−0.402+0.11−0.112+0.46−0.462+0.46−0.462+0.23−0.232+0.37−0.482+0.50−0.442+0.27−0.152+0.44−0.532+0.49−0.402+0.27−0.132+0.47−0.512+0.45−0.412+0.25−0.152+0.43−0.542+0.49−0.402+0.37−0.102+0.49−0.502+0.44−0.432+0.22−0.172+0.35−0.502+0.60−0.432+0.28−0.182+0.41−0.532+0.45−0.402+0.24−0.182+0.38−0.512+0.55−0.412+0.30−0.162+0.52−0.542+0.40−0.392+0.14−0.112=0.063,ClosenessRatioX1=ξX1=DX1,X∗DminX1,X∗−DX1,X−DmaxX1,X−=0.0630.030−0.0440.076=1.483.Further comparative analysis is conducted to assess the impact of a different aggregating method for SF-TOPSIS-SWGM. The same computations are presented in Tables11–13. From Table 14, the ranking results obtained by the proposed SWGM operator are slightly different from those using the SWAM operator. More specifically, the ordering of the alternatives has also changed a little due to the different attitudes of DMs. (X3 > X8 > X7 > X6 > X5 > X1 > X2 > X9 > X10 > X4).Table 13
Aggregate the judgments of DMs (SF-TOPSIS-SWGM operator).
X1X2X3X4X5X6X7X8X9X10PR1(0.41, 0.61, 0.27)(0.50, 0.52, 0.32)(0.31, 0.69, 0.23)(0.68, 0.34, 0.29)(0.55, 0.50, 0.25)(0.41, 0.60, 0.33)(0.35, 0.67, 0.25)(0.25, 0.76, 0.09)(0.80, 0.20, 0.18)(0.70, 0.30, 0.28)PR2(0.67, 0.37, 0.22)(0.75, 0.26, 0.25)(0.38, 0.66, 0.25)(0.75, 0.26, 0.21)(0.48, 0.56, 0.17)(0.62, 0.45, 0.20)(0.68, 0.33, 0.31)(0.59, 0.42, 0.37)(0.80, 0.20, 0.18)(0.80, 0.20, 0.18)PR3(0.60, 0.42, 0.39)(0.65, 0.39, 0.28)(0.57, 0.45, 0.36)(0.75, 0.26, 0.23)(0.58, 0.45, 0.17)(0.57, 0.46, 0.21)(0.51, 0.50, 0.44)(0.68, 0.36, 0.30)(0.54, 0.47, 0.36)(0.60, 0.40, 0.37)PR4(0.54, 0.47, 0.41)(0.48, 0.53, 0.40)(0.69, 0.34, 0.28)(0.60, 0.41, 0.38)(0.58, 0.44, 0.31)(0.56, 0.46, 0.38)(0.39, 0.63, 0.29)(0.64, 0.40, 0.28)(0.78, 0.23, 0.22)(0.40, 0.60, 0.36)SQ1(0.66, 0.39, 0.25)(0.41, 0.61, 0.32)(0.59, 0.43, 0.37)(0.52, 0.51, 0.30)(0.49, 0.55, 0.29)(0.53, 0.51, 0.26)(0.36, 0.67, 0.19)(0.64, 0.39, 0.27)(0.72, 0.30, 0.28)(0.35, 0.66, 0.30)SQ2(0.56, 0.47, 0.35)(0.52, 0.50, 0.38)(0.61, 0.42, 0.29)(0.50, 0.520.39)(0.66, 0.39, 0.26)(0.51, 0.52, 0.28)(0.68, 0.35, 0.29)(0.52, 0.50, 0.25)(0.50, 0.53, 0.34)(0.38, 0.64, 0.29)SQ3(0.75, 0.28, 0.30)(0.59, 0.57, 0.62)(0.63, 0.49, 0.58)(0.56, 0.62, 0.70)(0.43, 0.46, 0.42)(0.38, 0.55, 0.45)(0.49, 0.41, 0.30)(0.33, 0.33, 0.30)(0.35, 0.19, 0.38)(0.10, 0.29, 0.28)SQ4(0.59, 0.44, 0.41)(0.43, 0.52, 0.64)(0.60, 0.55, 0.46)(0.48, 0.57, 0.59)(0.58, 0.52, 0.37)(0.42, 0.48, 0.56)(0.55, 0.44, 0.41)(0.34, 0.16, 0.37)(0.32, 0.36, 0.31)(0.34, 0.41, 0.38)SQ6(0.53, 0.49, 0.35)(0.63, 0.49, 0.45)(0.54, 0.60, 0.41)(0.55, 0.47, 0.70)(0.41, 0.55, 0.57)(0.48, 0.44, 0.61)(0.48, 0.56, 0.31)(0.27, 0.21, 0.30)(0.29, 0.28, 0.31)(0.29, 0.22, 0.29)IT1(0.49, 0.54, 0.32)(0.61, 0.58, 0.57)(0.65, 0.45, 0.56)(0.50, 0.36, 0.79)(0.44, 0.44, 0.46)(0.39, 0.59, 0.47)(0.53, 0.66, 0.22)(0.23, 0.35, 0.31)(0.28, 0.20, 0.30)(0.31, 0.16, 0.21)IT2(0.62, 0.41, 0.28)(0.57, 0.59, 0.53)(0.59, 0.46, 0.62)(0.48, 0.62, 0.74)(0.43, 0.45, 0.50)(0.44, 0.57, 0.41)(0.55, 0.40, 0.29)(0.40, 0.30, 0.31)(0.35, 0.17, 0.19)(0.25, 0.32, 0.25)IT4(0.60, 0.41, 0.39)(0.70, 0.61, 0.63)(0.64, 0.55, 0.62)(0.46, 0.72, 0.63)(0.35, 0.41, 0.40)(0.39, 0.49, 0.40)(0.57, 0.29, 0.38)(0.31, 0.37, 0.28)(0.31, 0.30, 0.33)(0.32, 0.27, 0.35)IT5(0.66, 0.37, 0.30)(0.58, 0.60, 0.62)(0.66, 0.57, 0.53)(0.61, 0.71, 0.37)(0.44, 0.41, 0.42)(0.35, 0.45, 0.51)(0.44, 0.30, 0.64)(0.42, 0.37, 0.32)(0.35, 0.30, 0.23)(0.26, 0.27, 0.24)LO1(0.39, 0.66, 0.61)(0.70, 0.48, 0.50)(0.56, 0.71, 0.63)(0.53, 0.63, 0.35)(0.41, 0.32, 0.55)(0.52, 0.47, 0.30)(0.38, 0.47, 0.25)(0.34, 0.33, 0.29)(0.26, 0.30, 0.36)(0.24, 0.35, 0.44)LO2(0.67, 0.53, 0.66)(0.62, 0.50, 0.52)(0.67, 0.41, 0.73)(0.60, 0.34, 0.49)(0.35, 0.39, 0.53)(0.50, 0.39, 0.62)(0.29, 0.40, 0.30)(0.41, 0.35, 0.37)(0.22, 0.35, 0.28)(0.24, 0.26, 0.37)LO3(0.48, 0.35, 0.69)(0.59, 0.52, 0.58)(0.67, 0.55, 0.73)(0.70, 0.54, 0.66)(0.32, 0.46, 0.49)(0.44, 0.36, 0.47)(0.30, 0.30, 0.33)(0.23, 0.31, 0.24)(0.40, 0.34, 0.28)(0.26, 0.27, 0.28)LO4(0.78, 0.22, 0.81)(0.66, 0.35, 0.71)(0.52, 0.70, 0.62)(0.71, 0.22, 0.78)(0.19, 0.35, 0.66)(0.29, 0.50, 0.30)(0.38, 0.29, 0.21)(0.09, 0.18, 0.30)(0.26, 0.27, 0.34)(0.28, 0.35, 0.27)Table 14
Distances to PIS, NIS, and closeness ratio of each alternative based on SWGM operator.
AlternativesD (Xi, X∗)D (Xi, X-)Closeness ratioRankX10.0770.0722.3966X20.0570.0881.5057X30.1130.0374.0891X40.0260.1070.12610X50.0750.0592.4365X60.0810.0612.6364X70.0930.0752.9993X80.1040.0513.6292X90.0390.1220.5048X100.0360.1150.4699max0.1130.122min0.0260.037Additionally, the calculation process of SF-TOPSIS based on the SWGM operator in respect with X1 alternative as follows:Distances to positive and negative ideal solutions based on the SWGM operator:(35)DX1,X−=12n∑i=1nαx1−αx−2+βx1−βx−2+γx1−γx−212∗170.27−0.162+0.67−0.792+0.26−0.112+0.45−0.252+0.49−0.712+0.23−0.242+0.40−0.392+0.52−0.552+0.37−0.182+0.36−0.262+0.56−0.682+0.39−0.282+0.44−0.242+0.51−0.712+0.24−0.192+0.37−0.252+0.56−0.692+0.34−0.282+0.50−0.332+0.44−0.622+0.29−0.202+0.34−0.262+0.54−0.602+0.39−0.172+0.35−0.312+0.58−0.632+0.34−0.222+0.33−0.242+0.62−0.712+0.30−0.172+0.41−0.312+0.52−0.642+0.27−0.172+0.40−0.312+0.52−0.642+0.37−0.312+0.44−0.252+0.49−0.702+0.29−0.232+0.26−0.262+0.68−0.682+0.24−0.242+0.45−0.272+0.48−0.682+0.29−0.242+0.32−0.232+0.62−0.712+0.32−0.222+0.52−0.152+0.41−0.812+0.21−0.102=0.072,DXi,X∗=12n∑i=1nαxi−αx∗2+βxi−βx∗2+γxi−γx∗212∗170.27−0.532+0.67−0.402+0.26−0.192+0.45−0.532+0.49−0.402+0.23−0.192+0.40−0.502+0.52−0.432+0.37−0.232+0.36−0.522+0.56−0.412+0.39−0.222+0.44−0.482+0.51−0.452+0.24−0.282+0.37−0.452+0.56−0.482+0.34−0.282+0.50−0.502+0.44−0.442+0.29−0.292+0.34−0.432+0.54−0.492+0.39−0.362+0.35−0.472+0.58−0.452+0.34−0.292+0.33−0.522+0.62−0.412+0.30−0.212+0.41−0.492+0.52−0.442+0.27−0.262+0.40−0.482+0.52−0.442+0.37−0.262+0.44−0.472+0.49−0.452+0.29−0.272+0.26−0.472+0.68−0.452+0.24−0.242+0.45−0.482+0.48−0.452+0.29−0.252+0.32−0.482+0.62−0.452+0.32−0.272+0.52−0.542+0.41−0.402+0.21−0.192=0.077,ClosenessRatioX1=ξX1=DX1,X∗DminX1,X∗−DX1,X−DmaxX1,X−=0.0770.026−0.0720.122=2.369.


## 4.5. Comparative Analysis

To assess the suggested model’s validity and robustness, we conduct a comparative analysis in this part utilizing the different spherical fuzzy aggregating operators. To do this, two SWAM and SWGM operators based spherical fuzzy sets, developed by Kutlu Gündoğdu and Kahraman [34], will be utilized. The formula of these two operators is given in equation (16)-(17), respectively. It is worth noting that Figure 6 contains the final results and rating of alternatives. As seen in Tables 10 and 14, while the closeness ratios of the various spherical fuzzy aggregating operators differ, the order of options remains constant. The difference in closeness ratios is due to the spherical fuzzy aggregating operators’ nature. Decision-makers may choose amongst them depending on their type, scope, and objectives.Figure 6
Radar plot of the closeness ratio, ranking based on the SWAM, SWGM operators.To be more specific, we aim at implementing a pairwise comparison of the rankings produced by the SF-TOPSIS models. This comparison is performed by estimating Spearman’s rho correlation ccoefficient. The Spearman’s rho correlation coefficient is calculated using the following equation:(36)R=1−6∑i=1ndi2nn2−1.Topping the alternative list is X3-Vietravel, X7-Saigon Tourist, and X8- BenThanh Tourist. The values of the Spearman’s rho correlation coefﬁcient of rankings and closeness ratio are 0.891 and 0.901, which are very high, respectively. These results revealed that the proposed SF-TOPSIS with different aggregating methods (SF-TOPSIS-SWAM and SF-TOPSIS-SWGM) performs very similarly, given in Figure6 and Table 15.Table 15
Pairwise correlation of the closeness ratio, ranking based on SWAM, SWGM operators.
Ranking SWGMCloseness ratio SWGMRanking SWAM0.891Closeness ratio SWAM0.901


## 4.6. Discussions

This study offered a novel SF-Delphi and SF-TOPSIS framework as an effective tool for examining 17 package tour criteria and ten travel operator alternatives in Vietnam. To ascertain package tours’ primary characteristics and features, we invited 12 experts to conduct a survey. They determined that four categories (price, service quality, information and technology, and location) are the most crucial components that tourists consider before choosing a package tour. Decision makers play an essential role in delivering knowledge about alternatives in decision-making challenges. Due to time restrictions, a lack of information, knowledge, or data, and specialists’ imperfect knowledge of the topic, information about criteria weights is typically and primarily unknown. Many strategies have been offered in the literature to circumvent this limitation. One of the practical techniques to determining the crucial criterion is the SF-Delphi technique. When the information about the criterion is incomplete, the relative preference weights of the criteria are considered using spherical fuzzy scales and calculated using the SWAM operator.Regarding [70] the relative importance weights’ results, our study also shows that price is the most essential factor consistent with the prior studies [2, 70]. Based on the relative importance of package tour attribute, the findings showed that transportation cost and package discount criteria that customers most value besides the tour cost and insurance option with regard to price dimensions. Package tour operators should be conscious of their consumers’ price sensitivity and strive to deliver an added, nonmonetary value [2]. Accordingly, customers also concerned about tour operator quality. Tourism and service organizations have fiercely competed to attract tourists and acquire market share and profit by improving service quality and marketing guidelines to persuade tourists to return to an area [44]. A good destination package product is optimized to include a variety of tourist-friendly aspects that align with desired tourist experiences, enhance offering features, and deliver overall value [2]. The necessity of creating meaningful experiences in terms of local culture is evident such as an attribute that serves as the foundation for package tour differentiation as a good business strategy for travel operators [2]. Similarly, our study also found that operators are putting a greater emphasis on improving the tourist experience through their participation. Tourists impact tourism decision making by influencing payment methods in the context of information and technology [2, 70].The ultimate ranking of alternatives is determined in this study using the SF-TOPSIS method's closeness ratios. We used similarity between alternative and positive and negative ideal solutions regardless of distance. It is clear that our proposed method has the advantage of dealing with MCDM problems with incomplete criteria weights. It suggested that X3-Vietravel was the most prominent travel company among the ten alternative package tour operators, followed by X7-Saigon Tourist and X8- BenThanh Tourist. These are the top three companies in the Top Ten inbound and outbound travel firm rankings in actual operations. Our findings were corroborated with the recent reports from Vienamcredit–Vietnam’s leading provider of business information (https://vietnamcredit.com.vn/) [93]. These companies were evaluated and ranked based on a variety of criteria, including their financial capacity as demonstrated by their most recent financial statements; their media reputation; a survey of tourists and industry experts; and an examination of each company’s capital, market, labor, revenue growth rate, profit, and operation plan.Additionally, the results in this survey are congruent with those in the Vietnam Tourism Industry Report [94]. It is devoted to studying and evaluating Vietnam’s tourism and hospitality sectors to assist businesses and investors in making informed decisions. It provides an in-depth examination of the current situation and potential trends in Vietnam’s tourist industry. The report is organized into five significant sections: The external environment’s (economic, social, technological, and legal) impact on the tourism sector, both globally and in Vietnam. The study contains data from the GSO, the General Department of Tourists, provincial and municipal tourism departments, the Ministry of Planning and Investment and international data sources such as the World Economic Forum and the United Nations World Tourism Organization (UNWTO).


## 5. Conclusions

### 5.1. Conclusions

This study proposes a novel approach integrating SF-Delphi and SF-TOPSIS under a Spherical Fuzzy environment to evaluate and select the appropriate package tour providers in the empirical case from Vietnam. By providing independent spaces of membership, nonmembership, and hesitancy degrees, the theory of spherical fuzzy sets enables a powerful tool for modeling the uncertainties in MCDM problems. First, a modified Delphi technique based on spherical fuzzy sets is constructed to validate the various criteria affecting the customers’ decision-making process. Then, the SF-TOPSIS method is applied to evaluate ten package tour operators considering four dimensions: price, service quality, information and technology, and location. The results highlighted that the customers and experts would prioritize the greatest priorities, including PR2, SQ4, IT4, and LO3 while choosing package tour products. By comparing and evaluating ten tour operators in Vietnam, the findings identified that the evaluation results were reliable, which can help deal with MCDM problems in the tourism and hospitality sectors. Furthermore, this article indicated some key findings to improve their tourism competitiveness according to the analysis results deduced by SF-Delphi and SF-TOPSIS approaches.


### 5.2. Limitations

While the proposed approach adds an operational value in the tourism and hospitality industry, this research still has some drawbacks. First, this study employs a hybrid of Delphi and Fuzzy TOPSIS approaches based on spherical fuzzy sets to pick package tours; however, additional MCDM techniques such as VIKOR, EDAS, COPRAS, and GRA may also be applied. Second, the proposed method does not take into account the potential interactions and relationships between the criteria. Additionally, the weights of criteria were not derived independently based on expert judgments of TOPSIS; therefore, the relative importance of each criterion can be affected. Finally, this proposed framework considered four dimensions and 21 subcriteria; however, with the increasing ambiguity of things and the complicacy of situations, additional work and research are required to define and establish more criteria, and some new decision-making environments are developed to meet practical needs.


### 5.3. Future Studies

To further explicate this conceptual process, research package tour selection by distributing broader survey questions to more prominent participants such as direct customers can indicate opportunities for future work. Combinations of the spherical fuzzy MCDM models, partial least squares-structural equation modeling (PLS-SEM) and artificial neural network (ANN) are advised in the following stage to emphasize the gap in the relative value of criteria related to package tour evaluation. Methodologies can examine package tour selection in various countries and other factors. This study shows how merging MCDM-based spherical fuzzy sets can result in a more successful evaluation model. Their findings can be compared to previous studies’ findings, which can be expanded in future studies by evaluating the relative performance of new MCDM techniques that utilize emerging current fuzzy extensions such as spherical fuzzy sets, intuitionistic fuzzy sets, and hesitant fuzzy sets.


## 5.1. Conclusions

This study proposes a novel approach integrating SF-Delphi and SF-TOPSIS under a Spherical Fuzzy environment to evaluate and select the appropriate package tour providers in the empirical case from Vietnam. By providing independent spaces of membership, nonmembership, and hesitancy degrees, the theory of spherical fuzzy sets enables a powerful tool for modeling the uncertainties in MCDM problems. First, a modified Delphi technique based on spherical fuzzy sets is constructed to validate the various criteria affecting the customers’ decision-making process. Then, the SF-TOPSIS method is applied to evaluate ten package tour operators considering four dimensions: price, service quality, information and technology, and location. The results highlighted that the customers and experts would prioritize the greatest priorities, including PR2, SQ4, IT4, and LO3 while choosing package tour products. By comparing and evaluating ten tour operators in Vietnam, the findings identified that the evaluation results were reliable, which can help deal with MCDM problems in the tourism and hospitality sectors. Furthermore, this article indicated some key findings to improve their tourism competitiveness according to the analysis results deduced by SF-Delphi and SF-TOPSIS approaches.


## 5.2. Limitations

While the proposed approach adds an operational value in the tourism and hospitality industry, this research still has some drawbacks. First, this study employs a hybrid of Delphi and Fuzzy TOPSIS approaches based on spherical fuzzy sets to pick package tours; however, additional MCDM techniques such as VIKOR, EDAS, COPRAS, and GRA may also be applied. Second, the proposed method does not take into account the potential interactions and relationships between the criteria. Additionally, the weights of criteria were not derived independently based on expert judgments of TOPSIS; therefore, the relative importance of each criterion can be affected. Finally, this proposed framework considered four dimensions and 21 subcriteria; however, with the increasing ambiguity of things and the complicacy of situations, additional work and research are required to define and establish more criteria, and some new decision-making environments are developed to meet practical needs.


## 5.3. Future Studies

To further explicate this conceptual process, research package tour selection by distributing broader survey questions to more prominent participants such as direct customers can indicate opportunities for future work. Combinations of the spherical fuzzy MCDM models, partial least squares-structural equation modeling (PLS-SEM) and artificial neural network (ANN) are advised in the following stage to emphasize the gap in the relative value of criteria related to package tour evaluation. Methodologies can examine package tour selection in various countries and other factors. This study shows how merging MCDM-based spherical fuzzy sets can result in a more successful evaluation model. Their findings can be compared to previous studies’ findings, which can be expanded in future studies by evaluating the relative performance of new MCDM techniques that utilize emerging current fuzzy extensions such as spherical fuzzy sets, intuitionistic fuzzy sets, and hesitant fuzzy sets.



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*Source: 4249079-2022-05-29.xml*