Comments from the author of “An object-parameter approach to predicting unknown data in incomplete fuzzy soft sets” [Appl. Math. Modell. 37 (2013) 4139–4146]

Accepted Manuscript Short communication Comments from the author of “An object-parameter approach to predicting unknown data in incomplete fuzzy soft sets” [Appl. Math. Modell. 37(2013) 4139-4146] Tingquan Deng, Yanmei Chen PII: DOI: Reference:

S0307-904X(15)00269-3 http://dx.doi.org/10.1016/j.apm.2015.03.065 APM 10562

To appear in:

Appl. Math. Modelling

Received Date: Revised Date: Accepted Date:

5 January 2015 8 February 2015 25 March 2015

Please cite this article as: T. Deng, Y. Chen, Comments from the author of “An object-parameter approach to predicting unknown data in incomplete fuzzy soft sets” [Appl. Math. Modell. 37(2013) 4139-4146], Appl. Math. Modelling (2015), doi: http://dx.doi.org/10.1016/j.apm.2015.03.065

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Comments from the author of “An object-parameter approach to predicting unknown data in incomplete fuzzy soft sets” [Appl. Math. Modell. 37(2013) 4139-4146] Tingquan Deng a b

a,1

, Yanmei Chen

b

College of Science, Harbin Engineering University, Harbin, 150001 P. R. China Department of Mathematics, Harbin Institute of Technology, Harbin, 150001 P. R. China

Abstract A shortcoming appeared in the publication of “An object-parameter approach to predicting unknown data in incomplete fuzzy soft sets” [Appl. Math. Modell. 37(2013) 4139-4146] is clarified and a cogent solution to that shortcoming is presented to make the original paper more comprehensive. Keywords: Soft set, Incomplete fuzzy soft set, Complete distance, Relative dominance degree In [1] an approach to predicting unknown entries in an incomplete fuzzy soft set was proposed. That method makes full use of the information between objects and parameters. Yang et al. [2] gave an example showing that the evaluation results obtained by Equations (3) and (7) in the original paper [1] may be greater than 1. We agree with Yang et al. and declare further that the evaluation results are possibly less than 0 as well due to the fact that the relative distances between two objects belonging to a parameter and between two parameters on an object are employed. The following example (Table 1) shows this fact. In Table 1, there are six unknown entries to be predicted. By employing (3) and (7) in the original paper, the evaluation results of six unknown entries are listed in Table 2 and Table 3, respectively. It is shown that some of the 1

Corresponding author, Email address: [email protected] (T.Q. Deng).

Preprint submitted to Applied Mathematical Modelling

April 23, 2015

Table 1: An incomplete fuzzy soft set (F , E)

U h1 h2 h3 h4

e1 e2 e3 e4 ∗ 0.2 0.1 0.4 0.3 ∗ 0.7 ∗ ∗ 0.6 0.5 0.3 0.5 0.8 ∗ ∗

Table 2: The evaluation results of unknown entries in Table 1 regarding objects

U h1 h2 h3 h4

e1 -0.2000

e2

e3

0.5333

e4 0.8500

0.0667 1.1000

0.8500

evaluation results fall in [0 , 1], some of them are greater than 1 and some of them less than 0. It is a fact that the example given by Yang et al. has been presented as Table 1 in the original paper. When computing unknown entries of that example, the authors have noticed that hobject = 1.2778 and hparameter = 21 21 1.3333. Neither of their weighted sum nor either of them is regarded as the corresponding degree. They represent merely a kind of scalar characteristic (some kind of ordering structure) of objects regarding parameters. In that example, when the evaluation value is greater than 1 or less than 0, a postprocessing method, i.e. a commonly used thresholding technique in information processing, has been introduced to cut it into [0 , 1]. The results have been displayed in pp. 4144 of [1]. It is the shortcoming of not explicitly indicating this postprocessing method in the original paper that puzzles readers. In this short communication, we clarifies this problem and provide a solution to it by revising Equation (3) in Proposition 4.4 and (7) in Proposition

2

Table 3: The evaluation results of unknown entries in Table 1 regarding parameters

U h1 h2 h3 h4

e1 -0.3500

e2

e3

e4

1.0000

1.7000

0.0500 0.7693

4.7, respectively, in the original paper as follows. P hobject = max(0 , min(1 , jl hparameter jl

0.8000

i∈Ul (hil

− dij )

)) |Ul | P k∈Ej (hjk − vkl ) = max(0 , min(1 , )) |Ej |

(1) (2)

It is clear that this solution is equivalent to thresholding the evaluation by (2) fall in by (1) and hobject results and that both of the resultant hobject jl jl [0 , 1]. The predicted value of unknown entry hjl obtained by the weighted sum of hobject and hobject falls surely in [0 , 1] and can be considered as the jl jl membership degree of the object hj belonging to the parameter el . Acknowledgements The authors would like to thank Yang et al. for spotting the shortcoming in [1] and the editor in chief of this journal for providing us a chance of response to them. This work was supported in part by the National Natural Science Foundation of China with grant 11471001. References [1] Tingquan Deng, Xiaofei Wang, An object-parameter approach to predicting unknown data in incomplete fuzzy soft sets, Appl. Math. Modell. 37(2013) 4139-4146. [2] Yong Yang, Juanping Song, Xindong Peng, Erratum to “An objectparameter approach to predicting unknown data in incomplete fuzzy soft sets” [Appl. Math.Modell. 37(2013) 4139-4146], Submitted to Appl. Math. Modell. 3