The aim of this paper is to make a proposal for a new extension of the MULTIMOORA method extended to deal with bipolar fuzzy sets. Bipolar fuzzy sets are proposed as an extension of classical fuzzy sets in order to enable solving a particular class of decision-making problems. Unlike other extensions of the fuzzy set of theory, bipolar fuzzy sets introduce a positive membership function, which denotes the satisfaction degree of the element x to the property corresponding to the bipolar-valued fuzzy set, and the negative membership function, which denotes the degree of the satisfaction of the element x to some implicit counter-property corresponding to the bipolar-valued fuzzy set. By using single-valued bipolar fuzzy numbers, the MULTIMOORA method can be more efficient for solving some specific problems whose solving requires assessment and prediction. The suitability of the proposed approach is presented through an example.

The management of very complex systems is the most complex, and therefore the most difficult task of the managers of today’s organizations. The effectiveness of the management and managers of an organization depends to a large extent on the quality of the decisions they make on a daily basis.

Decision-making and decisions are the core of managerial activities. Bearing in mind the globalization and, therefore, the dynamics of business doing, all of the above-stated have caused business and the decision-making process to become more demanding. Making quality decisions requires an ever more extensive preparation, which also involves the consideration of the different aspects of a decision, for the reason of which the decision-making process becomes considerably formalized. Thus, real problems and situations in real life are characterized by a large number of mostly conflicting criteria, whose strict optimization is generally impossible.

When it is necessary to make a decision on choosing one of several potential solutions to a problem, it is desirable to apply one of the models based on multiple-criteria decision-making methods (MCDM). This most often involves the process of selecting one of several alternative solutions, for which certain goals are set. When MCDM is concerned, Greco

So, all the problems of today are, in general, multi-criterial, primarily because problems are mainly related to the achievement of the objectives related to a larger number of, usually conflicting, criteria, which is a great approximation to real tasks in decision-making processes Das

Within MADM, some of the methods that have been proposed are: Weighted Sum Model (WSM) Fishburn (

The MULTIMOORA method Brauers and Zavadskas (

As noted above, the MULTIMOORA method was applied in order to solve a variety of problems, such as: using MULTIMOORA for ranking and selecting the best performance appraisal method Maghsoodi

However, most decisions made in the real world are made in an environment in which goals and constraints cannot be precisely expressed due to their complexity; therefore, a problem cannot be displayed exactly in crisp numbers Bellman and Zadeh (

In addition to the aforementioned extensions of the fuzzy set theory, Zhang (

Despite an advantage that can be achieved by using bipolar fuzzy logic, they are significantly less used for solving MCDM problems compared to other fuzzy logic extensions. The following examples can be mentioned as some of the really rare usages of BFS for solving MCDM problems: Alghamdi

It is also important to note that these are current researches. In addition, the bipolar logic has been considerably used in the neutrosophic set theory, where Uluçay

Therefore, in order to enable a wider use of the MULTIMOORA method for solving even a wider range of problems, a bipolar extension of the MULTIMOORA method is proposed in this paper. Accordingly, the paper is structured as follows: in Section

Let

Let

A single-valued bipolar fuzzy number (SVBFN)

Let

Let

Let

Let

Let

Let

Compared to the other MCDM methods, the MULTIMOORA method is characteristic because it combines three approaches, namely: the Ratio System (RS) Approach, the Reference Point (RP) Approach and the Full Multiplicative Form (FMF) Approach, in order to select the most appropriate alternative.

In addition, this method does not calculate and does not use the overall significance for ranking alternatives and selecting the most appropriate one. Instead of using an overall parameter for ranking alternatives, the final ranking order of the considered alternatives, as well as the selection of the most appropriate alternative, is based on the use of the theory of dominance.

For an MCDM problem that includes the m alternatives that should be evaluated on the basis of the n criteria, the computational procedure of the MULTIMOORA can be expressed as follows:

In particular case, when evaluation is made only on the basis of benefit criteria, Eq. (

overall significance,

maximal distance to the reference point, and

overall utility.

As a result of these rankings, the three different ranking lists are formed, representing the rankings based on the RS approach, the RP approach and the FMF approach of the MULTIMOORA method.

The final ranking of the alternatives is based on the dominance theory, i.e. the alternative with the highest number of appearances in the first positions on all ranking lists is the best-ranked alternative.

For an MCDM problem involving m alternatives and n criteria and K decision-makers, whereby the performances of the alternatives are expressed by using SVBFNs, the calculation procedure of the extended MULTIMOORA method can be expressed as follows:

In order to provide an easier evaluation, the following Likert scale, shown in Table

Nine-point Likert scale for expressing degree of satisfaction.

Satisfaction level | Numerical value |

Neutral/without attitude | 0 |

Extremely low | 1 |

Very low | 2 |

Low | 3 |

Medium low | 4 |

Medium | 5 |

Medium high | 6 |

High | 7 |

Very high | 8 |

Extremely high | 9 |

Absolute | 10 |

However, the respondents should be introduced that the values listed in Table

After forming initial decision-making matrix, obtained responses should be divided by 10 in order to transform it into the allowed interval

After calculating the group evaluation matrix and the group weights of the criteria, all the necessary prerequisites for applying all the three approaches integrated in the MULTIMOORA method are obtained. Based on the approach proposed by Stanujkic

It is evident that

This step can be explained through the following sub-steps:

In the case when evaluation is made only on the basis of benefit criteria, Eq. (

In this stage, the alternatives are ranked based on their overall importance, maximum distance to the reference point and overall utility. As a result of that, three ranking lists are formed.

Based on these ranking lists, the final ranking list of the alternatives is formed on the basis of the theory of dominance, i.e. the alternative with the largest number of appearances on the first position in the three ranking lists is the most acceptable.

In this section, a numerical example of purchasing rental space is considered in order to explain the proposed approach in detail.

Suppose that a company is planning to start its sales business in a new location, and therefore is looking for a new sales building. After the initial consideration of the available alternatives, four alternatives have been identified as suitable. For this reason, a team of three decision-makers (DMs) was formed with the aim of evaluating suitable alternatives based on the following criteria:

As previously reasoned, in this evaluation the ratings of the alternatives in relation to the criteria are expressed by using SVBFNs.

The ratings obtained from the first of the three DMs are shown in Table

The ratings obtained from the first of the three DMs.

7 | −2 | 7 | −3 | 5 | −1 | 7 | −5 | 8 | −1 | |

4 | −1 | 5 | −2 | 4 | −2 | 4 | −6 | 7 | −1 | |

7 | −1 | 3 | −1 | 2 | 0 | 2 | −1 | 7 | −2 | |

9 | −1 | 4 | −1 | 3 | 0 | 3 | −1 | 6 | −1 |

The ratings obtained from the first of the three DMs, in the form of SVBFNs.

The ratings obtained from the second and the third of the three DMs are accounted for in Table

The ratings obtained from the second of the three DMs, in the form of SVBFNs.

The ratings obtained from the third of the three DMs, in the form of SVBFNs.

The group decision matrix, calculated by applying Eq. (

The group decision-making matrix.

The weights obtained from the first of the three DMs by applying the PIPRECIA method Stanujkic

The weights of the criteria obtained from the first of the three DMs.

1 | 1 | 0.19 | ||

1.2 | 0.80 | 1.25 | 0.23 | |

0.9 | 1.10 | 1.14 | 0.21 | |

0.7 | 1.30 | 0.87 | 0.16 | |

1.2 | 0.80 | 1.09 | 0.20 | |

5.00 | 5.35 | 1.00 |

The group criteria weights.

0.19 | 0.17 | 0.19 | 0.18 | |

0.23 | 0.24 | 0.23 | 0.24 | |

0.21 | 0.22 | 0.21 | 0.21 | |

0.16 | 0.17 | 0.16 | 0.16 | |

0.20 | 0.21 | 0.20 | 0.21 | |

1.00 |

On the basis of the ratings from Table

The overall significances, accounted for in Table

The overall significances of the considered alternatives.

0.94 | 0.99 | -0.05 | 3 | |||

0.90 | 0.99 | -0.09 | 4 | |||

0.68 | 0.63 | 0.05 | 2 | |||

0.97 | 0.62 | 0.35 | 1 |

After that, the reference point shown in Table

The reference points.

1.00 | −0.20 | 0.29 | −0.02 |

The maximum distances to the reference point accounted for in Table

The ratings of the alternatives obtained based on the reference point approach.

0.08 | 0.16 | 0.13 | 0.29 | 0.10 | 0.08 | 4 | |

0.17 | 0.28 | 0.00 | 0.29 | 0.09 | 0.00 | 1 | |

0.13 | 0.33 | 0.38 | 0.05 | 0.06 | 0.05 | 3 | |

0.04 | 0.14 | 0.36 | 0.11 | 0.00 | 0.00 | 1 |

The overall utility shown in Table

The overall utility of the considered alternatives.

0.94 | 0.99 | −0.05 | 3 | |||

0.90 | 0.99 | −0.09 | 4 | |||

0.68 | 0.63 | 0.05 | 2 | |||

0.97 | 0.62 | 0.35 | 1 |

Finally, on the basis of the ranking orders shown in Tables

The final ranking order of the considered alternatives.

3 | 4 | 3 | 3 | |

4 | 1 | 4 | 4 | |

2 | 3 | 2 | 2 | |

1 | 1 | 1 | 1 |

As can be seen from Table

The bipolar fuzzy sets introduced two membership functions, namely the membership function to a set and the membership function to a complementary set.

On the other hand, the MULTIMOORA method is an efficient and already proven multiple-criteria decision-making method, which has been used for solving a number of different decision-making problems so far.

Therefore, an extension of the MULTIMOORA method enabling the use of single-valued bipolar fuzzy numbers is proposed in this article. The usability and efficiency of the proposed extension is successfully demonstrated on the example of the problem of the best location selection.

In the literature, numerous extensions of the MULTIMOORA methods have been proposed with the aim to adapt it for the use of grey system theory, fuzzy set theory, as well as various extensions of fuzzy set theory. Some extensions that enable the use of neutrosophic sets are also proposed. The mentioned extensions aim to exploit the specificities of particular sets for solving certain types of decision-making problems, and thus enable more efficient decision making.

Because of the specificity that bipolar fuzzy sets provide, the proposed expanded MULTIMOORA method can be expected to be acceptable for solving a particular class of complex decision-making problems.