In this paper, we presented another form of eight similarity measures between PFSs based on the cosine function between PFSs by considering the degree of positive membership, degree of neutral membership, degree of negative membership and degree of refusal membership in PFSs. Then, we applied these weighted cosine function similarity measures between PFSs to strategic decision making. Finally, an illustrative example for selecting the optimal production strategy is given to demonstrate the efficiency of the similarity measures for strategic decision making problem.

The similarity measures are important and useful tools for determining the degree of similarity between two objects. Measures of similarity between fuzzy sets have gained attention from researchers for their wide applications in various fields, such as pattern recognition, machine learning, decision making and image processing, many measures of similarity between fuzzy sets have been proposed and researched in recent years (see, Bustince

Recently, Cuong (

Although Atanassov’s intuitionistic fuzzy set theory and similarity measures have been successfully applied in different areas (see Tang

In the following, we introduce some basic concepts related to intuitionistic fuzzy sets and some similarity measure between IFSs.

An IFS is given by

For each IFS

Suppose that there are two IFSs:

Ye (

On the other hand, Tian (

In the following, we introduced the weighted cosine and cotangent similarity measures between IFSs and, respectively (see Ye,

Although Atanassov’s intuitionistic fuzzy set theory (see Atanassov,

A picture fuzzy set (PFS)

Let

Suppose that there are two PFSs:

A cosine similarity measure between PIFSs and is proposed as follows:

It is obvious that the proposition is true according to the cosine value.

It is obvious that the proposition is true.

When

In the following, we shall investigate the distance measure of the angle as

Obviously,

For any

For three vectors

If we consider the weights of

Obviously, the weighted cosine similarity measure of two PFSs

Similar to the previous proof method, we can prove the above three properties.

When the four terms like degree of positive membership, degree of neutral membership, degree of negative membership and degree of refusal membership are considered in PFSs, we further propose the cosine similarity measure and weighted cosine similarity measure between PFSs as follows:

Based on the cosine function, in this section, we shall propose two cosine similarity measures between PFSs and analyse their properties.

Let

When the four terms like degree of positive membership, degree of neutral membership, degree of negative membership and degree of refusal membership are considered in PFSs, we further propose two cosine similarity measures between PFSs as follows:

(1) Since the value of the cosine function is within

(2) For two PFSs

If

(3) Proof is straightforward.

(4) If

In many situations, the weight of the elements

By using similar proof in Proposition

In this section, we shall propose a cotangent similarity measures between PFSs as follows:

In this section, the cosine similarity measures for PFSs are applied to strategic decision making problems (adapted from Wei and Merigó,

Now, we consider another kind of unknown production strategy

The data on production strategies.

(0.53,0.33,0.09) | (1.00,0.00,0.00) | (0.91,0.03,0.02) | (0.85,0.09,0.05) | (0.90,0.05,0.02) | |

(0.89,0.08,0.03) | (0.13,0.64,0.21) | (0.07,0.09,0.05) | (0.74,0.16,0.10) | (0.68,0.08,0.21) | |

(0.42,0.35,0.18) | (0.03,0.82,0.13) | (0.04,0.85,0.10) | (0.02,0.89,0.05) | (0.05,0.87,0.06) | |

(0.08,0.89,0.02) | (0.73,0.15,0.08) | (0.68,0.26,0.06) | (0.08,0.84,0.06) | (0.13,0.75,0.09) | |

(0.33,0.51,0.12) | (0.52,0.31,0.16) | (0.15,0.76,0.07) | (0.16,0.71,0.05) | (0.15,0.73,0.08) | |

(0.17,0.53,0.13) | (0.51,0.24,0.21) | (0.31,0.39,0.25) | (1.00,0.00,0.00) | (0.91,0.03,0.05) |

The similarity measures between

Similarity measures | ||||

0.813 | 0.656 | 0.787 | ||

0.810 | 0.656 | 0.638 | ||

0.813 | 0.765 | 0.762 | ||

0.840 | 0.765 | 0.831 | ||

0.813 | 0.765 | 0.709 | ||

0.813 | 0.757 | 0.707 | ||

0.486 | 0.442 | 0.469 | ||

0.486 | 0.442 | 0.440 |

From the above numerical results in Table

In this paper, we presented another form of eight similarity measures between PFSs based on the cosine function between PFSs by considering the degree of positive membership, degree of neutral membership, degree of negative membership and degree of refusal membership in PFSs. Then, we applied these weighted cosine function similarity measures between PFSs to strategic decision making problem. Finally, an illustrative example for selection of the optimal production strategy is given to demonstrate the efficiency of the similarity measures for strategic decision making problem. In the future, the application of the proposed cosine similarity measures of PFSs needs to be explored in complex group decision making, risk analysis and many other fields under uncertain environments, such as dual hesitant fuzzy linguistic sets, dual hesitant fuzzy uncertain linguistic sets, interval-valued dual hesitant fuzzy linguistic sets, and so on (see Wei

The work was supported by the National Natural Science Foundation of China under Grant Nos. 61174149 and 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (No. 15YJCZH138) and the construction plan of scientific research innovation team for colleges and universities in Sichuan Province (15TD0004).