The aim of this manuscript is to propose a new extension of the MULTIMOORA method adapted for usage with a neutrosophic set. By using single valued neutrosophic sets, the MULTIMOORA method can be more efficient for solving complex problems whose solving requires assessment and prediction, i.e. those problems associated with inaccurate and unreliable data. The suitability of the proposed approach is presented through an example.

The MULTIMOORA (Multi-Objective Optimization by a Ratio Analysis plus the Full Multiplicative Form) was proposed by Brauers and Zavadskas (

The ordinary MULTIMOORA method has been proposed for usage with crisp numbers. In order to enable its use in solving a larger number of complex decision-making problems, several extensions have been proposed, out of which only the most prominent are mentioned: Brauers

The MULTIMOORA method has been applied for the purpose of solving a wide range of problems.

As some of the most cited, the studies that consider different problems in economics (Brauers and Zavadskas,

As some of the newest studies in which the MULTIMOORA method is used for solving various decision-making problems, the following ones can be mentioned: material selection (Hafezalkotob and Hafezalkotob,

A significant approach in solving complex decision-making problems was formed by adapting the multiple criteria decision-making methods for the purpose of using fuzzy numbers, proposed by Zadeh in the fuzzy set theory (Zadeh,

Based on the fuzzy set theory, some extensions are also proposed, such as: interval-valued fuzzy sets (Turksen,

In addition to the membership function proposed in fuzzy sets, Atanassov (

The intuitionistic fuzzy set is composed of membership (the so-called truth-membership)

In intuitionistic fuzzy sets, indeterminacy

Such a proposed neutrosophic set is composed of three independent membership functions named the truth-membership

Smarandache (

Compared with the fuzzy set and its extensions, the single valued neutrosophic set can be identified as more flexible, for which reason an extension of the MULTIMOORA method adapted for the purpose of using the single valued neutrosophic set is proposed in this approach.

Therefore, the rest of this paper is organized as follows: in Section

Let

Let

For an SVNS

Let

Let

Let

Let

Let

The MULTIMOORA method consists of three approaches named as follows: the Ratio System (RS) Approach, the Reference Point (RP) Approach and the Full Multiplicative Form (FMF).

The considered alternatives are ranked based on all three approaches and the final ranking order and the final decision is made based on the theory of dominance. In other words, the alternative with the highest number of appearances in the first positions on all ranking lists is the best-ranked alternative.

In this approach, the compared alternatives are ranked based on

As in the RSA, the compared alternatives are ranked based on their

For an MCDM problem involving

For the sake of simplicity,

In order to demonstrate the applicability and efficiency of the proposed approach, an example has been adopted from Stanujkic

Suppose that a mining and smelting company has to build a new flotation plant, for which reason an expert has been engaged to evaluate the three Comminution Circuit Designs (CCDs) listed below:

The ratings obtained from the expert are shown in Table

The ratings of the three generic CCDs obtained from an expert.

The ranking orders of the alternatives obtained on the basis of the RS approach.

Rank | ||||||

0.425 | 0.045 | 0.380 | 2 | |||

0.372 | 0.006 | 0.366 | 3 | |||

0.583 | −0.263 | 0.845 | 1 |

The reference point.

The maximum distances from each alternative to the coordinate

The ranking order of the alternatives obtained based on the RP approach.

I | II | III | IV | V | VI | VII | VI |

Rank | |||||||

0.02 | 0.03 | 0.00 | 0.00 | 0.00 | 0.034 | 1 | |

0.05 | 0.02 | 0.02 | 0.00 | 0.01 | 0.048 | 2 | |

0.00 | 0.00 | 0.00 | 0.06 | 0.01 | 0.063 | 3 |

The ranking order of the alternatives obtained based on the FMF.

Rank | ||||||

0.618 | 0.498 | 1.242 | 3 | |||

0.605 | 0.481 | 1.258 | 2 | |||

0.674 | 0.283 | 2.379 | 1 |

The final ranking order of the alternatives which summarizes the three different ranks provided by the respective parts of the MULTIMOORA method is shown in Table

The final ranking order of the alternatives according to the MULTIMOORA method.

RS | RP | FMF | Rank | |

2 | 1 | 3 | 3 | |

3 | 2 | 2 | 2 | |

1 | 3 | 1 | 1 |

As it can be seen from Table

The MULTIMOORA method has been proven in solving different decision-making problems. In order to enable its application in the solving of a larger number of complex decision-making problems, numerous extensions have been proposed for the MULTIMOORA method.

Compared to crisp, fuzzy, interval-valued and intuitionistic fuzzy numbers, the neutrosophic set provides significantly greater flexibility, which can be conducive to solving decision-making problems associated with uncertainty, estimations and predictions.

Therefore, an extension of the MULTIMOORA method enabling the use of single valued neutrosophic numbers is proposed in this paper.

The usability and efficiency of the proposed extension is presented in the example of the comminution circuit design selection.

Finally, it should be noted that the proposed extension of the MULTIMOORA method can be used for solving a much larger number of complex decision-making problems. A number of real-world decision making problems which have to be solved is based on the data acquired from respondents can be identified as one of the areas where the proposed extension of the MULTIMOORA method can reach its advantages.