An extension of the Integrated Simple Weighted Sum Product (WISP) method is presented in this article, customized for the application of single-valued neutrosophic numbers. The extension is suggested to take advantage that the application of neutrosophic sets provides in terms of solving complex decision-making problems, as well as decision-making problems associated with assessments, prediction uncertainty, imprecision, and so on. In addition, an adapted questionnaire and appropriate linguistic variables are also proposed in the article to enable a simpler and more precise collection of respondents’ attitudes using single-valued neutrosophic numbers. An approach for deneutrosophication, i.e. the transformation of a single-valued neutrosophic number into a crisp number is also proposed in the article. Detailed use and characteristics of the presented improvement are shown on an example of the evaluation of rural tourist tours.

According to numerous similar definitions, multiple criteria decision-making (MCDM) is a process of evaluating or ranking alternatives based on a set of mutually conflicting criteria (Levy,

Since the end of the last century, MCDM has been used for solving many decision-making problems, and as a result, numerous MCDM methods have been proposed, such as SAW (MacCrimon,

In addition to the mentioned MCDM methods, a significant emergence of newly proposed MCDM methods can also be observed, such as ARAS (Zavadskas and Turskis,

However, it should be noted that the emergence of fuzzy sets, introduced by Zadeh (

In order to solve the decision-making problems associated with uncertainties and predictions, many MCDM methods have been extended to allow the use of fuzzy numbers. However, the use of only one membership function did not allow solving some types of complex decision-making problems, which is why certain extensions of the fuzzy set theory were proposed, as, for example, interval-valued fuzzy sets (Turksen,

In intuitionistic fuzzy set theory, Atanassov (

The membership and non-membership functions in relation to an intuitionistic fuzzy set are also known as truth-membership and falsity-membership functions.

In 1998, Smarandache (

To facilitate usage of neutrosophic sets for solving scientific and engineering problems, Wang

So far, numerous studies have been conducted to apply SVN sets for solving decision-making problems (Garg,

Stanujkic

Therefore, the remaining sections are subject to the subsequent organization. In Section

Some fundamental elements about neutrosophic sets, important for the development of the proposed extension, are presented in this section. In addition, some other contents that are also important for the development of the proposed extension are also discussed in this section.

Let

If

For an SVN set

Let

Let

Let

Let

Let

Let

The use of SVN numbers for collecting respondents’ attitudes also requires the use of a specially designed questionnaire. It has already been stated that SVN numbers use three membership functions, which allows the use of complex evaluation criteria. Instead of using ordinary questionnaires based on questions prepared for collecting ratings of alternatives concerning the selected criteria, the proposed questionnaire uses affirmative sentences whose truthfulness should be assessed using three membership functions.

For example, the first criterion, used in numerical illustration, Destination attractiveness, integrates the natural attractions of a tourist destination, such as natural beauties, mountain ranges, lakes, rivers, landscapes, environmental protection, diversity of flora and fauna, and others. Using the three affiliation functions provided by neutrosophic numbers, the respondent can express the level of his agreement with the confirmatory sentence, the level of his disagreement, and the level of his uncertainty regarding the statements included in the confirmatory sentence. It should be noted that the evaluation based on the use of SVN numbers does not require the mandatory use of all three membership functions for evaluation. Depending on their opinions, respondents can use three, two, or even one membership function. In cases when one or two membership functions are not used in the evaluation the values of unused membership functions are automatically set to value 0.

Finally, using Eq. (

Linguistic variables are often used in various extensions of grey, fuzzy, intuitionistic fuzzy, and neutrosophic extensions of MCDM methods to facilitate and enable decision-makers, i.e. respondents, to more accurately evaluate alternatives.

For the purposes of this research, the following nine-point scale, shown in Table

Linguistic variables for expressing confidence levels.

Linguistic variable | Abbreviation | Crisp numerical value | Permissible value range |

Extremely High | EH | 9 | |

Very High | VH | 8 | |

High | H | 7 | |

Moderate High | MH | 6 | |

Moderate | M | 5 | |

Moderate Low | ML | 4 | |

Low | L | 3 | |

Very Low | VL | 2 | |

Extremely Low | EL | 1 |

In addition to the use of linguistic variables, i.e. their abbreviations, respondents can express their attitudes using the recommended crisp numerical values. However, if they want it or it is necessary, the respondents can express their attitudes more precisely using numbers from the interval

Similar to defuzzification in fuzzy sets, in the neutrosophy, a deneutrosophication is the process of transforming information contained in neutrosophic numbers into crisp values.

The transformation of neutrosophic information into a crisp value can be easily performed using Eq. (

In the case when all three coefficients tend to a value of one,

The Simple WISP method was proposed in Stanujkic

Denominators used in Eq. (

Deneutrosophication can be performed using Eq. (

Using the approach presented above, decision-makers can take advantage of the previously discussed benefits that SVN sets provide when gathering respondents’ attitudes. Also, using Eq. (

In order to give a demonstration of the applicability of the presented extension of the WISP procedure, one example of selecting a tourist destination for Nature & Rural Tourism was considered.

After considering alternatives from Serbia, Montenegro, Albania, Bulgaria, and Romania, it is determined that the demonstration was carried out on the example of choosing a tourist tour of Natural and Rural Tourism in Romania. One of the main reasons for choosing Romania was a wealth of useful information regarding tourist tours, including a wealth of photographs that enchant the natural beauties of the Transylvania region located in central Romania.

To attest the applicability of the proposed extension of the WISP method, an example of evaluation of a tourist destination, that is evaluation of rural tourist tours, is discussed in this section. The evaluation of several below mentioned alternatives was performed according to the following criteria:

The evaluation criteria were selected based on the criteria proposed by Ryglova

The meaning of the above criteria can be described as follows: the criterion Destination attractiveness includes the presence of natural attractions, mountain ranges, lakes, rivers, landscapes, beauties of untouched nature, diversity of flora and fauna, and so on. The criterion Additional facilities include facilities such as hiking, climbing, visiting man-made facilities such as castles and fortifications, cultural and social attractions, and so on, while the criterion Accommodation and comfort include the type of accommodation and additional amenities such as the Internet, Wi-Fi, television, and so on. The criterion Transportation and accessibility includes the way of arriving at the starting points of the tourist tour from the residence of the respondents. Finally, since the considered tours have different durations, the Price criterion is considered as a complex criterion that includes the price on a daily basis and the total price of the tourist tour.

In this research, the following rural tourist tours were selected for evaluation:

Wildlife Tour in Romania,

Family Tour of Romania,

Maramures and Bucovina Tour, and

4-Day Carpathian Trek: Bucegi Mountains and Piatra Craiului National Park,

Village Life in Transylvanian Carpathian Mountains, and

14 Days Full Donau Delta, Braşov and Apuseni Tour.

Information regarding the above rural tourist tours is available on the following websites:

The evaluation of alternatives, i.e. checking the usability and efficacy of the proposed procedure, was done on a small number of respondents. More precisely, the examination was performed on a sample of fifteen examinees. From the collected questionnaires, one characteristic was chosen to show in detail the steps of the proposed calculation procedure.

The completed questionnaire with the attitudes of the selected respondents, filled in with the combined use of linguistic variables and numbers, is shown in Table

The questionnaire obtained from the selected respondent.

6.5 | EL | 1.5 | M | EL | 5.5 | ML | 1.5 | M | 2.5 | ||||||

VH | VH | EL | MH | M | VL | 0.5 | 6.5 | ||||||||

MH | EL | EH | EL | VL | ML | EL | VL | ML | L | MH | |||||

MH | 0.7 | VH | EL | EL | L | ML | ML | ML | L | M | |||||

EH | VH | EL | M | VL | EL | VH | ML | 0.5 | M | ||||||

EH | H | H | VL | VL | VH | ML | VH |

A transformed questionnaire with the attitudes of the selected respondent.

6.5 | 1 | 1.5 | 5 | 0 | 1 | 5.5 | 0 | 0 | 4 | 0 | 1.5 | 5 | 0 | 2.5 | |

8 | 0 | 0 | 8 | 0 | 1 | 6 | 0 | 0 | 5 | 0 | 0 | 2 | 0.5 | 6.5 | |

6 | 0 | 1 | 9 | 1 | 2 | 4 | 1 | 2 | 4 | 0 | 0 | 3 | 0 | 6 | |

6 | 0 | 0.7 | 8 | 1 | 1 | 3 | 4 | 4 | 4 | 0 | 0 | 3 | 0 | 5 | |

9 | 0 | 0 | 8 | 0 | 1 | 5 | 2 | 1 | 8 | 0 | 0 | 4 | 0.5 | 5 | |

9 | 0 | 0 | 7 | 0 | 0 | 7 | 2 | 2 | 8 | 0 | 0 | 4 | 0 | 8 |

A normalized intuitionistic decision matrix, generated utilizing Eq. (

A normalized intuitionistic decision-making matrix.

For further applying the proposed calculation procedure, the weights of the criteria are necessary after this step. In the observed case, the weights of criteria were defined by means of the PIPRECIA method. The weights calculated based on the attitudes of the selected respondent are shown in Table

The weights of the criteria.

0.23 | 0.18 | 0.20 | 0.18 | 0.22 |

After determining criteria weights, the sums and products of weighted normalized neutrosophic ratings of the alternatives were calculated, for beneficial and non-beneficial criteria, as presented in Table

Sums and products of weight-normalized neutrosophic ratings of alternatives achieved based on beneficial and non-beneficial criteria.

Deneutrosophied values of sums and products of the weighted normalized neutrosophic ratings are shown in Table

Deneutrosophied values of sums and products of weighted normalized neutrosophic ratings.

0.80 | 0.47 | 0.20 | 0.53 | 0.33 | −0.33 | 1.70 | 0.38 | |

0.76 | 0.29 | 0.24 | 0.71 | 0.46 | −0.46 | 2.58 | 0.34 | |

0.79 | 0.45 | 0.21 | 0.55 | 0.35 | −0.35 | 1.77 | 0.37 | |

0.81 | 0.46 | 0.19 | 0.54 | 0.35 | −0.35 | 1.76 | 0.35 | |

0.74 | 0.27 | 0.26 | 0.73 | 0.47 | −0.47 | 2.72 | 0.36 | |

0.73 | 0.41 | 0.27 | 0.59 | 0.32 | −0.32 | 1.77 | 0.46 |

The values of four utility measures

The recalculated values of four utility measures

Recalculated values of four utility measures, overall utility measures, and ranking order of alternatives.

Rank | ||||||

0.91 | 0.98 | 0.73 | 0.94 | 0.889 | 5 | |

1.00 | 0.78 | 0.96 | 0.92 | 0.916 | 2 | |

0.92 | 0.96 | 0.75 | 0.94 | 0.889 | 4 | |

0.92 | 0.95 | 0.74 | 0.92 | 0.884 | 6 | |

1.00 | 0.78 | 1.00 | 0.93 | 0.927 | 1 | |

0.90 | 1.00 | 0.74 | 1.00 | 0.910 | 3 |

As it can be concluded on the basis of data in Table

Similar evaluations, done again with the attitudes of the remaining respondents, showed that there were some differences in the ranks of considered alternatives, which was expected. However, it also emphasizes the need for developing a neutrosophic extension of the WISP method that can be used for group decision-making. Unfortunately, the development of such an extension has not yet been considered.

Numerous articles and studies dealing with the application of MCDM methods in the tourism and hospitality industry can be found in scientific and professional journals. A comprehensive overview of previously conducted research in this area can be found in Ahmad (

A similar approach to choosing a tourist destination was considered in Genç and Filipe (

An upgrading of the Simple WISP method based on the usage of single-valued neutrosophic numbers is proposed in this article.

The SVN numbers use three membership functions for expressing truth, indeterminacy, and falsity which is why they can be used for expressing beliefs, uncertainties, and doubts about some occurrences, conditions, or events. For this reason, these numbers can be very useful for collecting respondents’ attitudes because they provide respondents with a very flexible way of expressing attitudes. It is known that the three membership functions are mutually independent and that each function can have a value from the interval

The use of SVN numbers for collecting respondents’ attitudes also allows the use of complex criteria for evaluating alternatives. Of course, the use of these numbers requires the use of customized questionnaires, as well as adapted linguistic variables for expressing the respondents’ preferences, which are also discussed in the article.

Some initial research conducted during the development of the proposed approach pointed to certain problems related to the collection of attitudes from respondents who are not familiar with the use of neutrosophic sets. The questionnaire proposed in this article is certainly not suitable for collecting the views of respondents “on the street”, but can be used to collect the views of respondents who are familiar with the basic elements of fuzzy, intuitionistic, and neutrosophic sets.

Therefore, the intention to conduct a study with a significantly larger number of respondents using the proposed approach can be stated as one of the directions of future research. Adoption of the proposed approach for use in a group decision-making environment can also be mentioned as one of the further potential directions of research regarding the proposed approach.

An approach for deneutrosophication, i.e. the transformation of information contained in SVN numbers into crisp numbers is also considered in the article. Using this approach, decision-makers can analyse a variety of scenarios, from pessimistic to optimistic, similar as in many fuzzy and intuitionistic extensions of other MCDM methods.

Using the proposed approach, decision-makers can take advantage of the fact SVN sets provide for gathering respondents’ attitudes based on a smaller number of complex evaluation criteria. Also, using approach proposed for deneutrosophication, decision-makers can vary the impact of truth, indeterminacy, and falsity membership functions and consider different scenarios, from very pessimistic to very optimistic. And finally, the evaluations performed with the proposed extension of the Simple WISP method confirmed its applicability and efficiency.

Besides the outlined usefulness of the proposed approach, it could not be denied that it has some limitations, as well. Maybe the crucial shortcoming of the proposed approach reflects in its complexity for application by ordinary decision-makers that are not familiar with neutrosophic sets logic. In that sense, its application is limited only to those decision-makers who understand and successfully work with this type of decision-making aiding technique.

The results of conducted research proved the reliability and usability of the proposed extension, so it is considered that it would be also an adequate decision-making aid in other business fields such as project management, human resource management, production management, and so on. Besides, the recommendation of the future work involves the proposing of the extension of the WISP method based on the multi-valued neutrosophic numbers to acknowledge the vagueness and uncertainty of the environment to a greater extent.