Soft Covering Based Rough Sets and Their Application

Soft rough sets which are a hybrid model combining rough sets with soft sets are defined by using soft rough approximation operators. Soft rough sets can be seen as a generalized rough set model based on soft sets. The present paper aims to combine the covering soft set with rough set, which gives rise to the new kind of soft rough sets. Based on the covering soft sets, we establish soft covering approximation space and soft covering rough approximation operators and present their basic properties. We show that a new type of the soft covering upper approximation operator is smaller than soft upper approximation operator. Also we present an example in medicine which aims to find the patients with high prostate cancer risk. Our data are 78 patients from Selçuk University Meram Medicine Faculty.


Introduction
We can not solve the problems by using mathematical tools generally in the social life since in mathematics the concepts are precise and not subjective. Some theories were developed to eliminate this lack of vagueness such as fuzzy set theory [1], rough set theory [2], and soft set theory [3].
The fuzzy set theory initiated by Zadeh [1] in 1965 provides a useful framework for modelling and manipulating vague concepts. The fuzzy set theory is based on fuzzy membership function. Fuzzy membership function determines the belongness of an element to a set to a degree. Since being established, this theory has been actively studied by both mathematicians and computer scientists.
The rough set theory [2] proposed by Pawlak in 1982 is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information. The classical rough set theory is based on equivalence relations, but it was extended to covering based rough sets [4,5]. It is well known that the topology and rough set theory have been applied in many science and engineering areas such as chemistry, biology, crystal, image processing, knowledge acquisition, pattern recognition, engineering control, and biomedicine.
In 1999, Molodtsov [3] proposed the concept of a soft set, which can be seen as a new mathematical approach to vagueness. The absence of any restrictions on the approximate description in soft set theory makes this theory very convenient and easily applicable in practice. Maji et al. [6] carried out Molodtsov's idea by introducing several operations in soft set theory. Ali et al. [7] introduced some operations over soft sets. After that, Ge and Yang [8] further investigated these operational rules in [6,7] and obtained some interesting results including some different viewpoints with [7]. Ge et al. [9] established some relations between topology and soft set theory. Aktaş and Ç agman [10] compared soft sets to the related concepts of fuzzy sets and rough sets, providing examples to clarify their differences.
Feng et al. [11] investigated the concept of soft rough set in 2010 which is a combination of soft and rough sets. In [11,12], basic properties of soft rough approximations were presented and supported by some illustrative examples. In fact, a soft set instead of an equivalence relation was used to granulate the universe of discourse.
In this paper, we investigate the concept of soft covering based rough set which is a combination of covering soft set and rough set. We establish a soft covering approximation space. We supply an example to show that the new type of 2 The Scientific World Journal the soft rough set which is based on covering soft set is more accurate than soft rough set. On the other hand, we find that a new type of the soft covering upper approximation operator is smaller than soft upper approximation operator. Feng [13] gave an application of soft rough approximations in multicriteria group decision making problems and his method enables us to select the optimal object in more reliable manner. In this work, we use soft covering approximations at Feng's method and we present an example in medicine which aims to obtain the optimal choice for applying biopsy to the patients with prostate cancer risk.
Prostate cancer is the second most common cause of cancer death among men in most industrialized countries. It depends on various factors as family's cancer history, age, ethnic background, and the level of prostate specific antigen (PSA) in the blood. Since PSA is a substance produced by the prostate, it is very important factor to an initial diagnosis for patients [14][15][16]. As known, when the prostate cancer can be diagnosed earlier, the patient can be completely treated. The definitive diagnosis of the prostate cancer is possible with prostate biopsy. The results of PSA test, rectal examination, and transrectal findings help the doctor to decide whether biopsy is necessary or not [17][18][19]. If there is a biopsy for diagnosing, the cancer may spread to the other vital organs [17]. For this reason, the biopsy method is undesirable. In this study, we aim to reduce the number of patients who applied biopsy. Therefore, we give a new method which determines the necessity of biopsy and it gives to user a range of the risk of the cancer. For this process, it is used as laboratory data, prostate specific antigen (PSA), free prostate specific antigen (fPSA), prostate volume (PV), and age of the patient. We observe that this method is more rapid, economical, and without risk than the traditional diagnostic methods.

Preliminaries
In this section, we introduce the fundamental ideas behind fuzzy sets, rough sets, soft sets, soft rough sets, and fuzzy soft sets. Throughout this paper, the universe is supposed to be a finite nonempty set, 0 the empty set, and − the complement of in .
Definition 1 (see [1]). Let be a universe set. A fuzzy set in is a set of ordered pairs: where : → [0, 1] = is a mapping and ( ) (or ( )) states the grade of belongness of in . The family of all fuzzy sets in is denoted by .
Definition 2 (see [2]). Let be finite set and an equivalence relation on . Then the pair ( , ) is called a Pawlak approximation space.
are called the positive, negative, and boundary regions of , respectively. Now, we are ready to give the definition of rough sets.
Proposition 4 (see [2]). Let ( , ) be a Pawlak approximation space and , ⊆ . The properties of Pawlak's rough sets are  Let be an initial universe set and let be the set of all possible parameters with respect to . Parameters are often attributes, characteristics, or properties of the objects in . Let ( ) denote the power set of . Then a soft set over is defined as follows.
In other words, a soft set over is a parameterized family of subsets of the universe . For ∀ ∈ , ( ) may be considered the set of -approximate elements of the soft set The Scientific World Journal 3 = ( , ). It is worth noting that ( ) may be arbitrary. Some of them may be empty and some may have nonempty intersection [3]. Example 6. Miss Zeynep and Mr. Ahmet are going to marry and they want to hire a wedding room. The soft set ( , ) describes the "capacity of the wedding room. " Let = { 1 , 2 , 3 , 4 , 5 , 6 } be the wedding rooms under consideration and Table 1).
Although rough sets and soft sets are two different mathematical tools for modelling vagueness, there are some interesting connections between them.
Theorem 7 (see [10]). Every rough set may be considered a soft set.
As pointed out by several researchers, information systems and soft sets are closely related [20,21]. Given a soft set = ( , ) over a universe . if and are both nonempty finite sets, then = ( , ) could induce an information system in a natural way. In fact, for any attribute ∈ , one can define a function : → = {0, 1} by Therefore, every soft set may be considered an information system. This justifies the tabular representation of soft sets used widely in the literature. Conversely, it is worth noting that a soft set can also be applied to express an information system. Let = ( , ) be an information system. Taking as the parameter set, then a soft set ( , ) can be defined by setting where ∈ and V ∈ . Maji et al. [22] defined the following hybrid model fuzzy soft sets, combining soft sets with fuzzy sets.
Example 9 (see [23]). Miss X and Mr. Y are going to marry and they want to hire a wedding room. The fuzzy soft set ( , ) describes the "capacity of the wedding room. " Let = { , , , , } be the wedding rooms under consideration, Feng et al. [11] investigated the concept of soft rough set in 2010 which is a combination of soft and rough sets. A soft set instead of an equivalence relation was used to granulate the universe of discourse. The result was deviation of Pawlak approximation space called a soft approximation space [11]. For more details on this topic, we refer the interested reader to [11,12]. All proofs can be found there.
Definition 10 (see [11]). Let = ( , ) be a soft set over . Then the pair = ( , ) is called a soft approximation space. Based on the approximation space , we define the following two operations: assigning to every subset ⊆ two sets apr ( ) and apr ( ), which are called the soft -lower approximation and the soft -upper approximation of , respectively. In general, we refer to apr ( ) and apr ( ) as soft rough approximations of with respect to .
In addition, are called the soft -positive, negative, and boundary regions of , respectively. If apr ( ) = apr ( ), is said to be soft -definable; otherwise, is called a soft -rough set.
Theorem 14 (see [12]). Let = ( , ) be a soft set over and = ( , ) a soft approximation space. Then the following conditions are equivalent: To show the relationship between soft rough sets and Pawlak's rough sets, we first observe that soft sets and binary relations are closely related [11,12].
Theorem 15 (see [11,12]). Let = ( , ) be a soft set over . Then = ( , ) induces a binary relation ⊆ × , which is defined by for all ∈ , ∈ . Conversely, assume that is a binary relation from to . Define a set valued mapping : → ( ) by where all ∈ . Then = ( , ) is a soft set over . Moreover, it is seen that = and = .
The following results show that Pawlak's rough set model can be viewed as a special case of soft rough sets [12].
Theorem 16 (see [12]). Let be an equivalence relation on , = ( , ) the canonical soft set of , and = ( , ) a soft approximation space. Then, for all ⊆ , where ( ) and ( ) are the Pawlak rough approximations of . Thus, in this case, ⊆ is a (Pawlak) rough set if and only if is a soft -rough set.

Soft Covering Based Rough Sets
In this section, we use a special kind of soft set with rough set and establish a soft covering approximation space and present its basic properties.
We indicate a covering soft set with . In order to describe an object, we need only the essential characteristics related to this object, not all characteristics for this object. That is the purpose of minimal description concept.
In addition, are called the soft covering positive, negative, and boundary regions of , respectively.
A soft rough set is based on a soft set in a soft approximation space, whereas a soft covering based rough set is based on a covering soft set in a soft covering approximation space. We can call the soft rough set which is given in [11] as the first type of soft covering based rough set in a soft covering approximation space. From the definitions of two types of soft covering lower approximation operations, it is easy to see that the new soft covering lower approximation is the same as that in the first type of soft covering based rough set model. Therefore, we can give the following results.
Thus, apr ( ) ⊊ * ( ).      The Scientific World Journal The following examples show that the equalities mentioned above do not hold.

Multicriteria Group Decision Making
Feng [13] applied soft rough sets to multicriteria group decision making problem. The soft rough set based decision making method in [13] can be summarized as follows.
Step 2. Construct the evaluation soft set 1 = ( , ) using the primary evaluation results of the expert group .
Step 4. Compute the corresponding fuzzy sets 1 ,
Step 6. Input the weighting vector and compute the weighted evaluation values ( ) of each alternative ∈ . Then rank all the alternatives according to their weighted evaluation values; one can select any of the objects with the largest weighted evaluation value as the most preferred alternative.
We use this method to help doctors in diagnosing the prostate cancer risk. In our work, we use soft covering approximations instead of soft rough approximations in Step 3. We may expect to gain much more useful information with the help of soft covering approximations.

An Application of Multicriteria Group Decision Making by New Type of Soft Covering Approximation Operators
Feng [13] gave an application of soft rough approximations in multicriteria group decision making problems and his method enables us to select the optimal object in more reliable manner. In this work, we used soft covering approximations at Feng's method and aim to obtain the optimal choice for applying biopsy to the patients with prostate cancer risk by using the PSA, fPSA, PV, and age data of patients. We determine the risk of prostate cancer. Our aim is to help the doctor to determine whether the patient needs biopsy or not. We choose 78 patients from Selçuk University Medicine Faculty with prostate complaint as the data (see Table 2).
When we rank all the alternatives according to their weighted evaluation values, we can select any of the objects with the largest weighted evaluation value as the highest cancer risk. The results are as follows: Our results show that 0,83 is the highest value and 46 patients have this value and the patients with the membership 0,83 are potential cancer patients and they are under the highest risk. They need biopsy exactly. Two patients with 0,75      Table 4: Tabular presentation of the fuzzy soft set = ( , ) with weighted evaluation value of several patients. The Scientific World Journal 9 value are also under middle risk and they should be followed up by the doctor. The other patients are under low risk and they do not need the biopsy. According to data from Selçuk University Medicine Faculty, the biopsy is applied to all 78 patients, but only 44 patients were diagnosed with cancer. That is, 34 patients do not need the biopsy. According to our study, we obtained that the biopsy is necessary only to a group of 46 patients who are under high cancer risk. This group also contains 44 patients who were diagnosed with cancer. Hence, we reduce the number of patients who need biopsy.