The extensions of ordinary fuzzy sets are problematic because they require decimal numbers for membership, non-membership and indecision degrees of an element from the experts, which cannot be easily determined. This will be more difficult when three or more digits’ membership degrees have to be assigned. Instead, proportional relations between the degrees of parameters of a fuzzy set extension will make it easier to determine the membership, non-membership, and indecision degrees. The objective of this paper is to present a simple but effective technique for determining these degrees with several decimal digits and to enable the expert to assign more stable values when asked at different time points. Some proportion-based models for the fuzzy sets extensions, intuitionistic fuzzy sets, Pythagorean fuzzy sets, picture fuzzy sets, and spherical fuzzy sets are proposed, including their arithmetic operations and aggregation operators. Proportional fuzzy sets require only the proportional relations between the parameters of the extensions of fuzzy sets. Their contribution is that these models will ease the use of fuzzy set extensions with the data better representing expert judgments. The imprecise definition of proportions is also incorporated into the given models. The application and comparative analyses result in that proportional fuzzy sets are easily applied to any problem and produce valid outcomes. Furthermore, proportional fuzzy sets clearly showed the role of the degree of indecision in the ranking of alternatives in binomial and trinomial fuzzy sets. In the considered car selection problem, it has been observed that there are minor changes in the ordering of intuitionistic and spherical fuzzy sets.

Several new extensions of ordinary fuzzy sets appear in the literature every year. Starting with type-2 fuzzy sets (Zadeh,

The similar problem has been handled by some researchers in different ways (Atanassov

An eagle.

Without using a meter, answering these questions is really hard. But we can state that the wing length of the eagle is about three times its tail’s length and the wing length of the eagle is about 1.5 times the length from the top of the eagle’s head to its feet. If we can state such proportions, then the problem can be solved by more correct and objective data than with the estimated metric measurements.

In this paper, proportional fuzzy sets (PFS) are developed to represent a vague and imprecise definition of membership parameters and to develop proportional fuzzy set extensions such as proportional intuitionistic fuzzy sets and proportional picture fuzzy sets. This new way to determine membership, non-membership and indecision degrees is easier and more correct than the direct assignment of a decimal membership degree. PFS hypothesis is that humans can better express their judgments by using proportional judgments instead of decimal membership degree judgments. For instance, let an expert assign a picture fuzzy number

In this paper, imprecise proportions such as “around 2.5” or “between 3.5 and 4” are also handled to show how to model them in arithmetic operations and aggregation operators. Triangular and trapezoidal membership functions of proportions are considered for the mentioned four fuzzy sets extensions.

The rest of the paper is organized as follows. Section

In this section, we introduce proportional fuzzy set extensions. We first present the basic equations for each PFS extension, then give their arithmetic operations and aggregation operators.

Consider the intuitionistic fuzzy set

For instance, consider the proposition “

It should be also indicated that

Thus, a proportional intuitionistic fuzzy set

Addition and multiplication operations are defined as in Eqs. (

Let

Let

Consider the Pythagorean fuzzy set

For instance, consider the proposition “

Thus the proportional intuitionistic fuzzy sets can be represented by Eq. (

Let

Assuming no refusal degree, the multiplication by a constant and power operation are given by Eqs. (

Let

Let

Consider the picture fuzzy set

For instance, consider the proposition “

Thus the proportional picture fuzzy sets can be represented by Eq. (

Let

The multiplication by a constant and power operation are given by Eqs. (

Let

Let

Consider the spherical fuzzy set

For instance, consider the proposition “

Thus the proportional spherical fuzzy sets (PSFS) can be represented by Eq. (

Let

When the refusal degree is equal to zero, then Eq. (

Assuming no refusal degree, the multiplication by a constant and power operation are given by Eqs. (

Let

Let

In this section, we show how the imprecise proportion definitions are incorporated into the developed PFS extensions.

Experts can predict the proportions as an imprecise term such that membership degree is “around 3 times” or “between 2 and 3 times” larger than the hesitancy degree. Figures

For triangular fuzzy proportion prediction:

We obtain the corresponding

When PPyFS are considered, Figs.

Using triangular

Using triangular

Figure

When PSFS are considered, Fig.

Using triangular

Figure

In this section, we give an MCDM application of the developed PFS extensions for both precise and imprecise definitions of proportions.

Three experts (E1, E2, and E3) evaluate four cars by considering 5 attributes, which are comfort, safety, esthetic, price, and service facilities. Each of the experts constructs his/her decision matrix as given in Table

Proportional fuzzy decision matrices of the experts.

Alternatives | Comfort | Safety | Esthetic | Price | Service facilities |

E-1 | Criteria | ||||

Car-1 | |||||

Car-2 | (Around 5, Around 2) | ||||

Car-3 | |||||

Car-4 | |||||

E-2 | Criteria | ||||

Car-1 | |||||

Car-2 | |||||

Car-3 | |||||

Car-4 | (Around 6, Around 2) | ||||

E-3 | Criteria | ||||

Car-1 | (Around 5, Around 2) | ||||

Car-2 | |||||

Car-3 | |||||

Car-4 |

Giving some values for

When proportional intuitionistic fuzzy sets are considered, PIFWA operator in Eq. (

PIFWA operator with proportional intuitionistic fuzzy sets.

Alternatives | Comfort | Safety | Esthetic | Price | Service facilities | |||||

Car-1 | 0.6632 | 0.1862 | 0.6026 | 0.6547 | 0.6390 | 0.5121 | 0.7545 | 0.1228 | 0.6822 | 0.1809 |

Car-2 | 0.6892 | 0.1701 | 0.7482 | 0.1511 | 0.6724 | 0.1638 | 0.6790 | 0.3540 | 0.6926 | 0.1749 |

Car-3 | 0.6606 | 0.2072 | 0.6685 | 0.4132 | 0.6173 | 0.3518 | 0.6533 | 0.5329 | 0.6215 | 0.4175 |

Car-4 | 0.6685 | 0.1657 | 0.6914 | 0.1860 | 0.6237 | 0.5467 | 0.6177 | 0.5554 | 0.5429 | 0.5217 |

When proportional Pythagorean fuzzy sets are considered, PPFWA operator in Eq. (

PPFWA operator with proportional Pythagorean fuzzy sets.

Alternatives | Comfort | Safety | Esthetic | Price | Service facilities | |||||

Car-1 | 0.9390 | 0.2638 | 0.8923 | 0.4086 | 0.9185 | 0.3600 | 0.9743 | 0.1591 | 0.9471 | 0.2518 |

Car-2 | 0.9492 | 0.2336 | 0.9696 | 0.1970 | 0.9452 | 0.2306 | 0.9402 | 0.3124 | 0.9517 | 0.2411 |

Car-3 | 0.9340 | 0.2927 | 0.9382 | 0.2837 | 0.9113 | 0.3308 | 0.9284 | 0.3061 | 0.9061 | 0.3883 |

Car-4 | 0.9432 | 0.2345 | 0.9503 | 0.2562 | 0.9068 | 0.3887 | 0.9026 | 0.3967 | 0.8378 | 0.5008 |

When proportional picture fuzzy sets are considered, PPiFWA operator in Eq. (

PPFWA operator with proportional picture fuzzy sets.

Alts. | Comfort | Safety | Esthetic | Price | Service facilities | ||||||||||

Car-1 | 0.663 | 0.146 | 0.186 | 0.603 | 0.113 | 0.281 | 0.639 | 0.109 | 0.251 | 0.754 | 0.123 | 0.123 | 0.682 | 0.132 | 0.181 |

Car-2 | 0.689 | 0.131 | 0.170 | 0.748 | 0.092 | 0.151 | 0.672 | 0.164 | 0.164 | 0.679 | 0.094 | 0.226 | 0.693 | 0.128 | 0.175 |

Car-3 | 0.661 | 0.123 | 0.207 | 0.669 | 0.120 | 0.202 | 0.617 | 0.153 | 0.224 | 0.653 | 0.116 | 0.216 | 0.621 | 0.111 | 0.266 |

Car-4 | 0.669 | 0.166 | 0.166 | 0.691 | 0.119 | 0.186 | 0.624 | 0.107 | 0.268 | 0.618 | 0.109 | 0.272 | 0.543 | 0.119 | 0.332 |

PSFWA operator with proportional spherical fuzzy sets.

Alts. | Comfort | Safety | Esthetic | Price | Service facilities | ||||||||||

Car-1 | 0.939 | 0.207 | 0.264 | 0.892 | 0.165 | 0.409 | 0.918 | 0.156 | 0.360 | 0.974 | 0.159 | 0.159 | 0.947 | 0.184 | 0.252 |

Car-2 | 0.949 | 0.180 | 0.234 | 0.970 | 0.120 | 0.197 | 0.945 | 0.231 | 0.231 | 0.940 | 0.130 | 0.312 | 0.952 | 0.176 | 0.241 |

Car-3 | 0.934 | 0.173 | 0.293 | 0.938 | 0.168 | 0.284 | 0.911 | 0.226 | 0.331 | 0.928 | 0.164 | 0.306 | 0.906 | 0.162 | 0.388 |

Car-4 | 0.943 | 0.235 | 0.235 | 0.950 | 0.163 | 0.256 | 0.907 | 0.156 | 0.389 | 0.903 | 0.159 | 0.397 | 0.838 | 0.179 | 0.501 |

When proportional spherical fuzzy sets are considered, PSFWA operator in Eq. (

The three experts evaluate the five criteria as in Table

Linguistic proportional fuzzy scale.

Linguistic terms ( |
PF values |

Certainly Low (CL) | |

Very Low (VL) | |

Low (L) | |

Below Average (BA) | |

Average (A) | |

Above Average (AA) | |

High (H) | |

Very High (VH) | |

Certainly High (CH) |

Criteria evaluation by the experts.

Comfort | Safety | Esthetic | Price | Service facilities | |

E-1 | VH | VH | AA | CH | AA |

E-2 | H | VH | H | VH | H |

E-3 | A | VH | BA | AA | VH |

In the solution of the considered MCDM problem, we only use proportional intuitionistic fuzzy sets and proportional spherical fuzzy sets because of space constraints. From Table

Intuitionistic fuzzy weights based on PIFWA operator.

Comfort | Safety | Esthetic | Price | Service facilities | |||||

0.6043 | 0.3007 | 0.7273 | 0.1818 | 0.5375 | 0.3682 | 0.6993 | 0.2017 | 0.6562 | 0.2507 |

Sphericalfuzzy weights based on PSFWA operator.

Comfort | Safety | Esthetic | Price | Service facilities | ||||||||||

0.8860 | 0.1315 | 0.4349 | 0.9631 | 0.1204 | 0.2408 | 0.8204 | 0.1341 | 0.5431 | 0.9490 | 0.1318 | 0.2747 | 0.9252 | 0.1280 | 0.3530 |

The weighted decision matrix using intuitionistic numbers in Table

Weighted decision matrix using IFN and PIFWA.

Alternatives | Comfort | Safety | Esthetic | Price | Service facilities | |||||

Car-1 | 0.4008 | 0.4310 | 0.4383 | 0.7175 | 0.3435 | 0.6918 | 0.5276 | 0.2997 | 0.4476 | 0.3862 |

Car-2 | 0.4165 | 0.4197 | 0.5442 | 0.3054 | 0.3614 | 0.4717 | 0.4748 | 0.4843 | 0.4545 | 0.3817 |

Car-3 | 0.3992 | 0.4456 | 0.4862 | 0.5199 | 0.3318 | 0.5904 | 0.4569 | 0.6271 | 0.4078 | 0.5635 |

Car-4 | 0.4040 | 0.4166 | 0.5028 | 0.3340 | 0.3352 | 0.7136 | 0.4319 | 0.6451 | 0.3562 | 0.6416 |

By using intuitionistic fuzzy addition operation, we obtain the score of each alternative based on the simple additive weighting (SAW) method as in Table

IF scores based on SAW method.

IF scores | Net membership |
Ranking | ||

Car-1 | 0.9423 | 0.0248 | 0.9176 | 2 |

Car-2 | 0.9513 | 0.0112 | 0.9402 | 1 |

Car-3 | 0.9337 | 0.0483 | 0.8853 | 4 |

Car-4 | 0.9280 | 0.0411 | 0.8869 | 3 |

The weighted decision matrix using spherical fuzzy numbers in Table

Weighted decision matrix using SFN and PSFWA.

Alts. | Comfort | Safety | Esthetic | Price | Service facilities | ||||||||||

Car-1 | 0.8320 | 0.2397 | 0.4956 | 0.8594 | 0.1932 | 0.4639 | 0.7535 | 0.3684 | 0.6216 | 0.9247 | 0.2033 | 0.5659 | 0.8762 | 0.2110 | 0.4244 |

Car-2 | 0.8410 | 0.2183 | 0.4831 | 0.9338 | 0.1653 | 0.3075 | 0.7755 | 0.2315 | 0.5766 | 0.8923 | 0.1789 | 0.6266 | 0.8805 | 0.2054 | 0.4189 |

Car-3 | 0.8276 | 0.2118 | 0.5085 | 0.9036 | 0.1989 | 0.3658 | 0.7476 | 0.3292 | 0.6100 | 0.8810 | 0.2042 | 0.6234 | 0.8383 | 0.1908 | 0.5066 |

Car-4 | 0.8357 | 0.2635 | 0.4835 | 0.9152 | 0.1956 | 0.3462 | 0.7440 | 0.3956 | 0.6336 | 0.8566 | 0.1973 | 0.6726 | 0.7751 | 0.1992 | 0.5867 |

By using spherical fuzzy addition operation, we obtain the score of each alternative based on the simple additive weighting (SAW) method as in Table

SF scores based on SAW method.

SF scores | Net membership |
Ranking | |||

Car-1 | 0.9994 | 0.0291 | 0.0343 | 0.9505 | 2 |

Car-2 | 0.9997 | 0.0204 | 0.0225 | 0.9670 | 1 |

Car-3 | 0.9992 | 0.0329 | 0.0358 | 0.9469 | 3 |

Car-4 | 0.9988 | 0.0405 | 0.0418 | 0.9367 | 4 |

This comparative analysis based on different fuzzy set extensions’ arithmetic operations and aggregation operators gives slightly different ranking results. IF-SAW gives the ranking Car 2 > Car 1 > Car 4 > Car 3 whereas SF-SAW gives the ranking Car 2 > Car 1 > Car 3 > Car 4. This difference comes from the hesitancy computation in IFS and SFS.

We presented several proportional fuzzy set extensions including PIFS, PPyFS, PPiFS, and PSFS. The main advantage of these proportional fuzzy set extensions is their ability to determine the membership, non-membership, and hesitancy degrees easily and correctly. We developed the arithmetic operations and aggregation operators of each proportional fuzzy set extension. We also presented

Experts cannot assign numbers with multiple decimal places for any membership degree when they directly try to assign it. The proposed proportional approaches could produce membership degrees with several decimal places. Car alternatives in the application section have been prioritized by using simple additive weighting method based on proportional intuitionistic fuzzy sets and proportional spherical fuzzy sets. A slight difference has been obtained in their rankings because of the differences in the theoretical structures of the fuzzy set extensions. Intuitionistic fuzzy sets require membership and non-membership degrees to be assigned whereas spherical fuzzy sets require hesitancy degree additionally.

The limitation of proportional fuzzy sets may be difficult to implement in cases where the degrees are independent, as in neutrosophic sets. Because in neutrosophic sets, each degree can take any value between 0 and 1, and the upper limit of the sum can be 3.

For further research, we suggest the developed proportional fuzzy sets to be employed in the extension of MCDM methods such as VIKOR, ELECTRE, WASPAS, MOORA, or COPRAS. We developed only four extensions of ordinary fuzzy sets. We developed only four proportional fuzzy extensions of ordinary fuzzy sets. The other extensions such as neutrosophic sets, fermatean fuzzy sets, q-rung orthopair fuzzy sets, or t-spherical fuzzy sets can be handled to develop their proportional fuzzy versions.