Advanced Aspects of Computational Intelligence and Applications of Fuzzy Logic and Soft ComputingView this Special Issue
Spherical Fuzzy Decision-Making Approach Integrating Delphi and TOPSIS for Package Tour Provider Selection
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.
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 . However, after the COVID-19 pandemic, travel needs are predicted to increase considerably after many countries’ extended quarantine and border-closed . 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 . 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 , 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 .
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 . Regarding the study of , 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 . According to , 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 . First-time travelers contribute largely to package tour customers while seasoned travelers are few . Fesenmaier et al.  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  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 . 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 ; 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: Section 2 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  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  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  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  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 , Räikkönen and Honkanen  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  was in contrast with Davies and Downward , 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.  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  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.  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  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 .
The Delphi method  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 . 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 . 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 . As a result, the hybrid strategy significantly reduced the time and expense associated with achieving consensus without distorting the outcomes . 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 . 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 . First, Zadeh  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  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  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  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 . Ali and Smarandache  proposed neutrosophic fuzzy sets to generalize intuitionistic fuzzy sets. Cường  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 .
Zoraghi et al.  evaluated and ranked service quality in the hotel business using a fuzzy MCDM model in the tourism and hospitality industry. Tseng  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  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 Table 1. 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 . 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  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 . Service quality (SQ): The quality of the service provided is always the determinant factor that significantly influences the customers’ decisions . Lin  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  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.  identified the importance of the tour leader role in the customers’ decision-making process. Supporting this idea, Chang  proved that tour-guide performance greatly influenced the tourists’ satisfaction and experience, especially shopping behavior, which can contribute directly to the company’s revenue . 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 , 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 . 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 . 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 . 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 . 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 .
3.1. Spherical Fuzzy Sets: Basic Operations and Definitions
Kutlu Gündoğdu and Kahraman  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).
Definition 1. SFS is presented as :where denotes a spherical fuzzy set of the universe .with , for each , , and denote for membership, non-membership, and hesitancy levels of to , respectively.
Definition 2. Let and be two SFSs. Some arithmetic operations of SFS are presented as follows.(i)Union(ii)Intersection(iii)Addition(iv)Multiplication(v)Multiplication by a scalar; (vi)Power of
Definition 3. For these SFSs and , the followings are valid under the condition .
Definition 4. Spherical weighted arithmetic mean (SWAM) concerning ; ; , SWAM is defined as follows.
Definition 5. Spherical weighted geometric mean (SWGM) concerning ; ; , SWGM is defined as follows.
3.2. Proposed Method
This research aims to propose the novel two-phased spherical fuzzy MCDM approach of SF-Delphi and SF-TOPSIS (Figure 2): 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 Table 2. The significance vector for each indicator is obtained using the SWGM operator  and is shown in equation below: Step 2: Differentiating from the previous Delphi method [52, 91], we propose to defuzzy the aggregated criteria score using (15), the score function. Afterward, the threshold is attained as to validate criteria from the initial set. If , criterion is valid, and if , criterion 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  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 Definitions 1 and 5. Construct aggregated spherical fuzzy decision matrix based on decision-makers’ opinions : Step 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 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). 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). Step 7: Calculating the distances between alternative . The SF-NIS and SF-PIS are calculated using equation (22-23), respectively. Step 8: Determining the maximum distance to the SF-NIS and minimum distance to the SF-PIS using equations (24) and (25), respectively. 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. 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.
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 (Table 3) 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.
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. Section 1 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 , 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.
Then, 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 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.
Additionally, the computation process of the SF-Delphi method respect with PR1 criterion is presented as follows:
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.
According to the 17 selected criteria from Table 6, 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.
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 Table 7 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).
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:
Weighted decision matrix based on the SWAM operator :
Weighted decision matrix based on the SWGM operator :
Score function values based on the SWAM operator :
Score function values based on the SWGM operator :
Distances to positive and negative ideal solutions based on the SWAM operator:
Further comparative analysis is conducted to assess the impact of a different aggregating method for SF-TOPSIS-SWGM. The same computations are presented in Tables 11–13. 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).
Additionally, 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: