A type of similarity measure for vague soft sets and its application to landmark preference

Abstract

A type of similarity measure between two vague soft sets, which contains subitem satisfying properties instead of single value parameters defined within a value range, is introduced. Some properties of the proposed similarity measure are studied. Eight detailed expression examples derived from the subitem are given. A real-world problem in handling landmark preference is solved based on this technique of similarity measure of vague soft sets. The results of an analysis of the landmark preference example are based on a comparison of the proposed similarity measure with the existing similarity measures, which is meant to demonstrate the advantages of the proposed approach.

Keywords

Vague soft sets vague sets soft sets similarity measure landmark preference

1 Introduction

The concept of vague soft sets, which are combinations of vague sets and soft sets, was first presented by Xu et al. [1] to represent and address uncertain information. Vague soft sets are similar to intuitionistic fuzzy soft sets, as proposed by Maji et al. [2, 3], and they act as a parameterized family of fuzzy sets. The difference is that fuzzy sets are vague sets [4] in vague soft sets; however, in intuitionistic fuzzy soft sets, the fuzzy sets are intuitionistic fuzzy sets, as introduced by Atanassov [5]. Although Bustince and Burillo [6] showed that vague sets and intuitionistic fuzzy sets are equivalent by basic definition, Lu and Ng [7] showed that vague sets facilitate significantly better analysis in terms of data relationships, incompleteness, and similarity measures. The notion of vague sets was introduced by Gau and Buehrer [4] in 1993, and it considers the degree of membership, non-membership and hesitation to express the state of uncertain information. Vague sets are regarded as a further generalization of fuzzy sets, which were originally proposed by Zadeh [8]. According to [9], however, Molodtsov argued that the fuzzy, vague and intuitionistic fuzzy set theories have their own difficulties in incompatibility with parameterization tools, and he then introduced the concept of soft sets. Later, studies on the combination of incompleteness, fuzzy, probabilistic and soft set theories [10 –13] have been done. Maji et al. [2 , 14] developed many notions of fuzzy soft sets and intuitionistic fuzzy soft sets. Based on the work of Maji et al., Majumdar and Samanta [15] defined generalized fuzzy soft sets. In 2010, Xu et al. [1] introduced the concept of vague soft sets as an extension of soft sets and vague sets.

Similarity measure is an important concept in vague sets and vague soft sets. Researchers have studied the topic of similarity measures between vague sets from various points of view, such as similarity measures based on distance measures [16], statistical point of view [17], cosine similarity measure [18], and similarity measure with parameters [19]. Some similarity measures for vague sets are entirely new and these include similarity measures based on transformation techniques [20], direct operation on some kind of membership functions [21, 22], and distance to the most fuzzy intuitionistic set [23]. Due to combinations of vague sets and soft sets, several recently proposed similarity measures of vague soft sets are based on the existing similarity measures of vague sets and soft sets, respectively. Majumdar and Samanta [24] defined a similarity measure of vague soft sets based on the similarity measure of vague sets proposed by Lu et al. in [25]. Wang and Qu [26] introduced a similarity measure between two vague soft sets based on the similarity measure of vague sets proposed by Li and Xu in [27]. Muthukumar and Sai Sundara Krishnan [28] introduced a similarity measure and a weighted similarity measure on intuitionistic fuzzy soft sets based on the similarity measure of soft sets proposed by Majumdar and Samanta in [29]. At present, from our point of view, the existing similarity measures can fall into the following two categories: one without parameters, and the other with one or two parameters. Because these parameters are only a single value within a defined value range [30], the similarity measures using these parameters are limited in overcoming the drawbacks that do not fit well in some counterintuitive cases [31, 32].

Similarity measure of vague sets and vague soft sets is widely used in fuzzy decision making, pattern recognition, machine learning, image processing, fuzzy reasoning, medical diagnosis, and so forth. In this paper, the similarity measure of vague soft sets is introduced into a new application of landmark preference, which is the issue of how to order the alternative landmarks and choose the optimal landmark. Landmarks are very useful as reference points and play a guiding role in the communication of route directions. In [33], Nothegger et al. constructed a specialized computational model to address landmark preference problems. In fact, because each landmark can be characterized under different feature properties, i.e., facade size, shape factor, shape deviation, RGB color, HSB color, visibility, cultural importance and identifiability [33], landmark preference involves comprehensively evaluating the significance of a landmark, indicating landmark preference is a multi-criteria decision-making (MCDM) problem. In many MCDM methods, the technique of similarity measure of vague soft sets is one of useful tools to address MCDM problems [24].

The main motivation and contributions of this work are summarized as follows:

The existing similarity measures have been studied from various points of view, but they are usually a single detailed expression. Therefore, this paper aims to a type of similarity measure that can derive different detailed expressions through a subitem.

The existing similarity measures with one or two parameters are limited in overcoming the drawbacks in some counterintuitive cases. Therefore, the proposed similarity measure uses a subitem that satisfies certain properties to replace the usage of single value parameters.

Because landmark preference is a MCDM problem, this paper introduces the technique of similarity measure of vague soft sets to address the landmark preference problems, and thus can avoid constructing and using a specialized computational model as in [33].

The rest of this paper is organized as follows: In Section 2, some preliminary definitions and results regarding vague soft sets and similarity measures of vague soft sets are given. In Section 3, a new type of similarity measure of vague soft sets is defined, and the properties of the proposed similarity measure have been studied, and then, landmark preference based on similarity measure of vague soft sets is described. Section 4 describes an application of the proposed similarity measure in landmark preference and then conducts a comparison analysis. Section 5 includes a discussion, and Section 6 concludes the paper.

2 Preliminaries

Definition 2.1. [4] For any element of the finite universe of discourse U, a vague set A in the universe U = {u₁, u₂, …, u_n} can be expressed as A = {(u_i, [t_A (u_i) , 1 - f_A (u_i)]) |u_i ∈ U}, i.e., A (u_i) = [t_A (u_i) , 1 - f_A (u_i)], where the functions t_A (u_i) : U → [0, 1] and f_A (u_i) : U → [0, 1] denote a degree of true membership and a degree of false membership of the element u_i ∈ U to the set A, respectively, with 0 ≤ t_A (u_i) + f_A (u_i) ≤1, ∀ u_i ∈ U.

A subinterval [t_A (u_i) , 1 - f_A (u_i)] is called the vague value, which means the degree of membership of u_i in the vague set A. A third function, given by π_A (u_i) =1 - t_A (u_i) - f_A (u_i) , ∀ u_i ∈ U, 0 ≤ π_A (u_i) ≤1, is called the hesitancy degree of an element of the vague set A.

Definition 2.2. [4] Let A, B be two vague sets of the universe U = {u₁, u₂, …, u_n}, then the union, intersection, complement and subset of vague sets are defined as follows: $\begin{matrix} A \cup B = {(u_{i}, [t_{A} (u_{i}) \lor t_{B} (u_{i}), (1 - f_{A} (u_{i})) \lor (1 - \\ f_{B} (u_{i}))]) | u_{i} \in U}, \end{matrix}$ $A \cap B = {(u_{i}, [t_{A} (u_{i}) \land t_{B} (u_{i}), (1 - f_{A} (u_{i})) \land (1 -$ $f_{B} (u_{i}))]) | u_{i} \in U},$ $A^{c} = {(u_{i}, [f_{A} (u_{i}), 1 - t_{A} (u_{i})]) | u_{i} \in U},$ $\begin{matrix} A \subseteq B iff t_{A} (u_{i}) \leq t_{B} (u_{i}) and 1 - f_{A} (u_{i}) \leq 1 - \\ f_{B} (u_{i}), \forall u_{i} \in U, \\ A = B iff A \subseteq B and B \subseteq A . \end{matrix}$

Definition 2.3. [9] Let U = {u₁, u₂, …, u_n} be an initial universe set and E = {e₁, e₂, …, e_m} be a set of parameters. Let P(U) denote the power set of U. Let A ⊂ E. A pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P (U).

Definition 2.4. [1] Let U = {u₁, u₂, …, u_n} be an initial universe set and E = {e₁, e₂, …, e_m} be a set of parameters. Let V(U) denote the collection of all vague subsets of U. Let A ⊂ E. A pair (F, A) is called a vague soft set (VSS in short) over U, where F is a mapping given by F : A → V (U).

If (F, E) = {F (e_i) |i = 1, 2, …, m} and (G, E) = {G (e_i) |i = 1, 2, …, m} are two VSSs over U = {u₁, u₂, …, u_n}, then the following two relations can be found in [24, 26]:

Definition 2.5. [24, 26] (F, E) is a vague soft subset of (G, E) if for all e ⊆ E, F(e) is included by G(e). This relation is denoted by (F, A) ⊆ (G, B), which implies t_{F(e_i)} (u_j) ≤ t_{G(e_i)} (u_j) and 1 - f_{F(e_i)} (u_j) ≤1 - f_{G(e_i)} (u_j) (i = 1, 2, …, m j = 1, 2, …, n).

Definition 2.6. [24, 26] (F, E) and (G, E) are said to be equal if for all e ⊆ E, F(e) is included by G(e) and G(e) is included by F(e). This relation is denoted by(F, A) = (G, B) , which implies t_{F(e_i)} (u_j) = t_{G(e_i)} (u_j) and 1 - f_{F(e_i)} (u_j) =1 - f_{G(e_i)} (u_j) (i = 1, 2, …, m j = 1, 2, …, n).

Definition 2.7. [26] Let U = {u₁, u₂, …, u_n} be an initial universe set and E = {e₁, e₂, …, e_m} be a set of parameters. Let S : VSS (U) ²→ [0,1] be a mapping. For (F, E) ∈ VSS (U) and (G, E) ∈ VSS (U) , S ((F, E) , (G, E)) is a measure of similarity between (F, E) and (G, E) if it satisfies the following properties: $\begin{matrix} (H 1) S ((F, E), (G, E)) = S ((G, E), (F, E)); \\ (H 2) S ((F, E), (G, E)) \in [0, 1]; \\ (H 3) S ((F, E), (G, E)) = 1 \Leftrightarrow (F, E) = (G, E); \\ (H 4) S ((F, E), (G, E)) = 0 \Leftrightarrow \forall e_{i} \in E, u_{j} \in U, \\ t_{F (e_{i})} (u_{j}) = 0, f_{F (e_{i})} (u_{j}) = 1, t_{G (e_{i})} (u_{j}) = 1, f_{G (e_{i})} (u_{j}) = 0, \\ or t_{F (e_{i})} (u_{j}) = 1, f_{F (e_{i})} (u_{j}) = 0, t_{G (e_{i})} (u_{j}) = 0, \\ f_{G (e_{i})} (u_{j}) = 1; \end{matrix}$ $\begin{matrix} (H 5) (F, E) \subseteq (G, E) \subseteq (P, E) \Rightarrow S ((F, E), (P, E)) \leq \\ S ((F, E), (G, E)) \land S ((G, E), (P, E)), where (P, E) \in \\ VSS (U) . \end{matrix}$

The properties defined in Definition 2.7 are the same as the ones presented in [24], except for the property (H4), which is regarded as a ‘strong’ version property.

Definition 2.8. [34] Let A, B be two vague sets of the universe U = {u₁, u₂, …, u_n}, then a Hausdorff distance d(A, B) is defined as follows: $\begin{matrix} d (A, B) = \frac{1}{n} \sum_{i = 1}^{n} \\ \max {| t_{A} (u_{i}) - t_{B} (u_{i}) |, | f_{A} (u_{i}) - f_{B} (u_{i}) |} . \end{matrix}$

3 Methodology

This section presents the similarity measure of vague soft sets proposed in this work and describes landmark preference based on the proposed similarity measure.

3.1 A type of similarity measure for vague soft sets

In [26], Wang and Qu inherited the idea of similarity measure between two vague sets from [27] and introduced a similarity measure between two VSSs, which concerns both degree of support and the differences between t_A and t_B, as well as that between f_A and f_B. Vague sets have the advantage of being able to consider hesitancy degree, but they have not yet considered it. Their definition of similarity measures are limited by reflecting on the use of degrees of belongingness and non-belongingness. Some existence of counter-intuitive cases results from the numerical comparisons of the similarity measure in [27] that does not seem reasonable [20, 23]. Due to the importance of hesitancy degree, Chen and Chang [20] analyzed the hesitancy degree and proposed a similarity measure based on transformation techniques. In their proposed similarity measure, an important item [π_A + π_B]/2 is presented. They used some examples to illustrate that the proposed similarity measure can outperform the existing similarity measures. Additionally, in [17 , 30], they presented several similarity measures with one or two parameters on vague sets, with parameters that are defined within a range of values. For example, the parameter p in [21] is defined as 1≤ p ≤ ∞. By using the parameters, these similarity measures can overcome some of the drawbacks of getting unreasonable results, but the effect is still limited because the parameters are only set as a single value within the defined value range. In this paper, a new type of similarity measure between two VSSs is proposed based on [26] and [20]. Instead of using such single value parameters defined within a value range, the proposed similarity measure contains a subitem that can derive different detailed expressions only if it satisfies three properties, as shown in the following definition:

Definition 3.1. Let S_ξ: VSS (U) ²→ [0,1] be a mapping, where VSS(U) denotes the set of all vague soft sets in the universe U = {u₁, u₂, …, u_n}. Let E = {e₁, e₂, …, e_m} be the universal set of parameters. If (F, E) ∈ VSS (U) and (G, E) ∈ VSS (U), then the similarity measure between (F, E) and (G, E) is defined by $S_{ξ} ((F, E), (G, E)) = \frac{\sum_{i = 1}^{m} M_{i} ((F, E), (G, E))}{m}$ (1)

with $\begin{matrix} M_{i} ((F, E), (G, E)) = \\ 1 - \frac{1}{4 n} \sum_{j = 1}^{n} [β_{1} ((F, E), (G, E)) + ξ ((F, E), (G, E)) \times \\ β_{2} ((F, E), (G, E)) \frac{π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j})}{2}], \end{matrix}$ $\begin{matrix} β_{1} ((F, E), (G, E)) = \\ [| S_{F (e_{i})} (u_{j}) - S_{G (e_{i})} (u_{j}) | + | t_{F (e_{i})} (u_{j}) - t_{G (e_{i})} (u_{j}) | + \\ | f_{F (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j}) |], \end{matrix}$ $\begin{matrix} β_{2} ((F, E), (G, E)) = \\ max (| t_{F (e_{i})} (u_{j}) - t_{G (e_{i})} (u_{j}) |, | f_{F (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j}) |), \end{matrix}$ $S_{F (e_{i})} (u_{j}) = t_{F (e_{i})} (u_{j}) - f_{F (e_{i})} (u_{j})$

and $S_{G (e_{i})} (u_{j}) = t_{G (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j}),$

where ξ ((F, E) , (G, E)) satisfies the following properties: $\begin{matrix} (R 1) ξ ((F, E), (G, E)) = ξ ((G, E), (F, E)); \\ (R 2) 0 \leq ξ ((F, E), (G, E)) \leq 1; \\ (R 3) (F, E) \subseteq (G, E) \subseteq (P, E) \Rightarrow ξ ((F, E), (G, E)) \leq \\ ξ ((F, E), (P, E)) and ξ ((G, E), (P, E)) \leq ξ ((F, E), \\ (P, E)), where (P, E) \in VSS (U) . \end{matrix}$

In Definition 3.1, ξ ((F, E) , (G, E)) is a subitem of S_ξ ((F, E) , (G, E)) that satisfies the three properties (R1), (R2) and (R3) rather than the parameters defined in [17 , 30], which obtain a single value within a defined value range. If ξ ((F, E) , (G, E)) =0, S_ξ ((F, E) , (G, E)) becomes the similarity measure introduced by Wang and Qu in [26].

β₂ ((F, E) , (G, E)) inherits the idea of Hausdorff distance between two vague sets, as defined in Definition 2.8. According to the properties of Hausdorff distance [34], it is easy to determine that β₂ ((F, E) , (G, E)) satisfies the following properties: $\begin{matrix} (T 1) 0 \leq β_{2} ((F, E), (G, E)) \leq 1; \\ (T 2) β_{2} ((F, E), (G, E)) = 0 iff (F, E) = (G, E); \\ (T 3) β_{2} ((F, E), (G, E)) = β_{2} ((G, E), (F, E)); \\ (T 4) If (F, E) \subseteq (G, E) \subseteq (P, E), (F, E), (G, E), \\ (P, E) \in VSS (U), then β_{2} ((F, E), (G, E)) \leq β_{2} ((F, E), \\ (P, E)) and β_{2} ((G, E), (P, E)) \leq β_{2} ((F, E), (P, E)) . \end{matrix}$

Proposition 3.1. The proposed similarity measure S_ξ ((F, E) , (G, E)) defined in Definition 3.1 satisfies all the properties given in Definition 2.7.

Proof. (H1) From [26], we have β₁ ((F, E) , (G, E)) = β₁ ((G, E) , (F, E)). From properties (R1) and (T3), we have M_i ((F, E) , (G, E)) = M_i ((G, E) , (F, E)) . Hence S_ξ ((F, E) , (G, E)) = S_ξ ((G, E) , (F, E)).

(H2) From properties (R2) and (T1), we have $\begin{matrix} 0 \leq ξ ((F, E), (G, E)) β_{2} ((F, E), (G, E)) \times \\ \frac{π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j})}{2} \leq π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j}) . \end{matrix}$

Considering 0 ≤ |S_{F(e_i)} (u_j) - S_{G(e_i)} (u_j) |≤2, we then obtain $\begin{matrix} 0 \leq β_{1} ((F, E), (G, E)) + ξ ((F, E), (G, E)) \times \\ β_{2} ((F, E), (G, E)) \frac{π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j})}{2} \\ \leq | S_{F (e_{i})} (u_{j}) - S_{G (e_{i})} (u_{j}) | + | t_{F (e_{i})} (u_{j}) - t_{G (e_{i})} (u_{j}) | \\ + | f_{F (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j}) | + π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j}) \\ \leq | S_{F (e_{i})} (u_{j}) - S_{G (e_{i})} (u_{j}) | + | t_{F (e_{i})} (u_{j}) + t_{G (e_{i})} (u_{j}) | \end{matrix}$

$\begin{matrix} \begin{matrix} + | f_{F (e_{i})} (u_{j}) + f_{G (e_{i})} (u_{j}) | + π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j}) \\ \leq 2 + 2 = 4, \end{matrix} \end{matrix}$

which implies M_i ((F, E) , (G, E)) ∈ [0, 1] and S_ξ ((F, E) , (G, E)) ∈ [0, 1].

(H3) From [26], we have $\begin{matrix} β_{1} ((F, E), (G, E)) = 0 \Leftrightarrow \forall e_{i} \in E, u_{j} \in U, t_{F (e_{i})} (u_{j}) = \\ t_{G (e_{i})} (u_{j}), f_{F (e_{i})} (u_{j}) = f_{G (e_{i})} (u_{j}) \Leftrightarrow (F, E) = (G, E) . \end{matrix}$

Let δ ((F, E) , (G, E)) = ξ ((F, E) , (G, E)) β₂ ((F, E) , $(G, E)) \times \frac{π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j})}{2} .$

From properties (T2), we then have $\begin{matrix} β_{1} ((F, E), (G, E)) + δ ((F, E), (G, E)) = 0 \Leftrightarrow \forall e_{i} \in E, \\ u_{j} \in U, t_{F (e_{i})} (u_{j}) = t_{G (e_{i})} (u_{j}), f_{F (e_{i})} (u_{j}) = f_{G (e_{i})} (u_{j}) \\ \Leftrightarrow (F, E) = (G, E) . \end{matrix}$

Hence

$\begin{matrix} S_{ξ} ((F, E), (G, E)) = 1 \Leftrightarrow \frac{\sum_{i = 1}^{m} M_{i} ((F, E), (G, E))}{m} = 1 \Leftrightarrow \\ M_{i} ((F, E), (G, E)) = 1 \Leftrightarrow β_{1} ((F, E), (G, E)) + δ ((F, E), \\ (G, E)) = 0 \Leftrightarrow \forall e_{i} \in E, u_{j} \in U, t_{F (e_{i})} (u_{j}) = t_{G (e_{i})} (u_{j}), \\ f_{F (e_{i})} (u_{j}) = f_{G (e_{i})} (u_{j}) \Leftrightarrow (F, E) = (G, E) . \end{matrix}$

(H4) Let δ ((F, E) , (G, E)) = ξ ((F, E) , (G, E)) $β_{2} ((F, E), (G, E)) \frac{π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j})}{2}$ and ρ ((F, E) , (G, E)) = |t_{F(e_i)} (u_j) - t_{G(e_i)} (u_j) | + |f_{F(e_i)} (u_j) - f_{G(e_i)} (u_j) | + δ ((F, E) , (G, E)) .

From the proof of (H2) above, we have

0 ≤ |S_{F(e_i)} (u_j) - S_{G(e_i)} (u_j) | ≤ 2 and 0 ≤ ρ ((F, E) , (G, E)) ≤ 2. Hence

$\begin{matrix} S_{ξ} ((F, E), (G, E)) = 0 \Leftrightarrow \frac{\sum_{i = 1}^{m} M_{i} ((F, E), (G, E))}{m} = 0 \\ \Leftrightarrow M_{i} ((F, E), (G, E)) = 0 \Leftrightarrow \frac{1}{4 n} \sum_{j = 1}^{n} [β_{1} ((F, E), (G, E)) \\ + δ ((F, E), (G, E))] = 1 \Leftrightarrow | S_{F (e_{i})} (u_{j}) - S_{G (e_{i})} (u_{j}) | = 2, \\ ρ ((F, E), (G, E)) = 2 \Leftrightarrow \forall e_{i} \in E, u_{j} \in U, t_{F (e_{i})} (u_{j}) = 0, \\ f_{F (e_{i})} (u_{j}) = 1, t_{G (e_{i})} (u_{j}) = 1, f_{G (e_{i})} (u_{j}) = 0, or t_{F (e_{i})} (u_{j}) \\ = 1, f_{F (e_{i})} (u_{j}) = 0, t_{G (e_{i})} (u_{j}) = 0, f_{G (e_{i})} (u_{j}) = 1 . \end{matrix}$

(H5) Let $θ ((F, E), (G, E)) = \frac{1}{2} ξ ((F, E), (G, E)) β_{2}$ ((F, E) , (G, E)) . From properties (R3) and (T4), we have (F, E) ⊆ (G, E) ⊆ (P, E) ⇒ θ ((F, E) , (G, E)) ≤ θ ((F, E) , (P, E)) , θ ((G, E) , (P, E)) ≤ θ ((F, E) , (P, E)) .

From (F, E) ⊆ (G, E) ⊆ (P, E) ⇒ t_{F(e_i)} (u_j) ≤ t_{G(e_i)} (u_j) ≤ t_{P(e_i)} (u_j) , f_{F(e_i)} (u_j) ≥ f_{G(e_i)} (u_j) ≥ f_{P(e_i)} (u_j) ⇒ S_{F(e_i)} (u_j) ≤ S_{G(e_i)} (u_j) ≤ S_{P(e_i)} (u_j) and 0 ≤ θ ((F, E) , (G, E)) ≤1/2,

we then have

$\begin{matrix} d ((F, E), (G, E)) \\ \begin{matrix} = \end{matrix} | S_{F (e_{i})} (u_{j}) - S_{G (e_{i})} (u_{j}) | + | t_{F (e_{i})} (u_{j}) - t_{G (e_{i})} (u_{j}) | + \\ | f_{F (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j}) | + θ ((F, E), (G, E)) \times \\ [π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j})] \\ \begin{matrix} = \end{matrix} | S_{G (e_{i})} (u_{j}) - S_{F (e_{i})} (u_{j}) | + | t_{G (e_{i})} (u_{j}) - t_{F (e_{i})} (u_{j}) | + \\ | f_{F (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j}) | + θ ((F, E), (G, E)) \times \\ [π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j})] \\ \begin{matrix} = \end{matrix} S_{G (e_{i})} (u_{j}) - S_{F (e_{i})} (u_{j}) + t_{G (e_{i})} (u_{j}) - t_{F (e_{i})} (u_{j}) + \\ f_{F (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j}) + θ ((F, E), (G, E)) \times \\ [π_{F (e_{i})} (u_{j}) + π_{G (e_{i})} (u_{j})] \\ \begin{matrix} = \end{matrix} t_{G (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j}) - t_{F (e_{i})} (u_{j}) + f_{F (e_{i})} (u_{j}) + \\ t_{G (e_{i})} (u_{j}) - t_{F (e_{i})} (u_{j}) + f_{F (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j}) + \\ θ ((F, E), (G, E)) [2 - t_{F (e_{i})} (u_{j}) - f_{F (e_{i})} (u_{j}) - \\ t_{G (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j})] \\ \begin{matrix} = \end{matrix} 2 [t_{G (e_{i})} (u_{j}) - t_{F (e_{i})} (u_{j}) + f_{F (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j})] + \\ θ ((F, E), (G, E)) [2 - t_{F (e_{i})} (u_{j}) - f_{F (e_{i})} (u_{j}) - \\ t_{G (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j})] \end{matrix}$

and

$\begin{matrix} d ((F, E), (P, E)) \\ = 2 [t_{P (e_{i})} (u_{j}) - t_{F (e_{i})} (u_{j}) + f_{F (e_{i})} (u_{j}) - f_{P (e_{i})} (u_{j})] + \\ θ ((F, E), (P, E)) [2 - t_{F (e_{i})} (u_{j}) - f_{F (e_{i})} (u_{j}) - \\ t_{P (e_{i})} (u_{j}) - f_{P (e_{i})} (u_{j})] . \end{matrix}$

Hence $\begin{matrix} d ((F, E), (G, E)) - d ((F, E), (P, E)) \\ = 2 [t_{G (e_{i})} (u_{j}) - t_{P (e_{i})} (u_{j}) + f_{P (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j})] + \\ θ ((F, E), (G, E)) [2 - t_{F (e_{i})} (u_{j}) - f_{F (e_{i})} (u_{j}) - \\ t_{G (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j})] - θ ((F, E), (P, E)) [2 - \\ t_{F (e_{i})} (u_{j}) - f_{F (e_{i})} (u_{j}) - t_{P (e_{i})} (u_{j}) - f_{P (e_{i})} (u_{j})] \\ \leq 2 [t_{G (e_{i})} (u_{j}) - t_{P (e_{i})} (u_{j}) + f_{P (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j})] + \\ θ ((F, E), (P, E)) [2 - t_{F (e_{i})} (u_{j}) - f_{F (e_{i})} (u_{j}) - \\ t_{G (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j})] - θ ((F, E), (P, E)) [2 - \\ t_{F (e_{i})} (u_{j}) - f_{F (e_{i})} (u_{j}) - t_{P (e_{i})} (u_{j}) - f_{P (e_{i})} (u_{j})] \\ = 2 [t_{G (e_{i})} (u_{j}) - t_{P (e_{i})} (u_{j}) + f_{P (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j})] + \\ θ ((F, E), (P, E)) [t_{P (e_{i})} (u_{j}) + f_{P (e_{i})} (u_{j}) - t_{G (e_{i})} (u_{j}) \\ - f_{G (e_{i})} (u_{j})] \\ = [2 - θ ((F, E), (P, E))] [t_{G (e_{i})} (u_{j}) - t_{P (e_{i})} (u_{j})] + \\ [2 + θ ((F, E), (P, E))] [f_{P (e_{i})} (u_{j}) - f_{G (e_{i})} (u_{j})] \\ \leq 0, \end{matrix}$

which implies $\begin{matrix} M_{i} ((F, E), (P, E)) \leq M_{i} ((F, E), (G, E)) \Rightarrow S_{ξ} ((F, E), \\ (P, E)) \leq S_{ξ} ((F, E), (G, E)) . \end{matrix}$

Similarly, we can prove S_ξ ((F, E) , (P, E)) ≤ S_ξ ((G, E) , (P, E)). Hence, we get S_ξ ((F, E) , (P, E)) ≤ min(S_ξ ((F, E) , (G, E)) , S_ξ ((G, E) , (P, E))) .

This completes the proof of Proposition 3.1.

Proposition 3.2. Let VSS(U) denotes the set of all vague soft sets in the universe U = {u₁, u₂, …, u_n}. Let (F, E) ∈ VSS (U) and (G, E) ∈ VSS (U). The subitem $ξ_{L} ((F, E), (G, E)) = \frac{max (φ ((F, E)), ψ ((G, E))) + 1}{2}$ satisfies the properties (R1), (R2) and (R3) given in Definition 3.1 if φ ((F, E)) and ψ ((G, E)) satisfy the following properties:

\begin{matrix} (C 1) 0 \leq φ ((F, E)) \leq 1, 0 \leq ψ ((G, E)) \leq 1; \\ (C 2) If (F, E) \subseteq (G, E) \subseteq (P, E), (P, E) \in VSS (U) \\ then φ ((F, E)) \leq ψ ((G, E)), φ ((G, E)) \leq ψ ((P, E)) \\ and φ ((F, E)) \leq ψ ((P, E)); \\ (C 3) If (F, E) \subseteq (G, E) \\ \subseteq (P, E), (P, E) \in VSS (U) then ψ ((F, E)) \leq ψ ((G, E)), \\ ψ ((G, E)) \leq ψ ((P, E)) and ψ ((F, E)) \leq ψ ((P, E)) . \end{matrix}

Proof. (R1) Obviously, we have

max(φ ((F, E)) , ψ ((G, E))) = max(ψ ((G, E)) , φ ((F, E))) , which implies ξ_L ((F, E) , (G, E)) = ξ_L ((G, E) , (F, E)).

(R2) From (C1), we have $\begin{matrix} \frac{1}{2} = \frac{max (0, 0) + 1}{2} \leq ξ_{L} ((F, E), (G, E)) = \\ \frac{max (φ ((F, E)), ψ ((G, E))) + 1}{2} \leq \frac{max (1, 1) + 1}{2} = 1 . \end{matrix}$

(R3) From (C2), we have $(F, E) \subseteq (G, E), φ ((F, E)) \leq ψ ((G, E)) \Rightarrow$ $\begin{matrix} ξ_{L} ((F, E), (G, E)) = \frac{max (φ ((F, E)), ψ ((G, E))) + 1}{2} \\ = \frac{ψ ((G, E)) + 1}{2} \end{matrix}$

and $\begin{matrix} (F, E) \subseteq (P, E), φ ((F, E)) \leq ψ ((P, E)) \Rightarrow \\ ξ_{L} ((F, E), (P, E)) = \frac{max (φ ((F, E)), ψ ((P, E))) + 1}{2} \end{matrix}$ $= \frac{ψ ((P, E)) + 1}{2} .$

From (C3), we then have ξ_L ((F, E) , (G, E)) ≤ ξ_L ((F, E) , (P, E)) . Similarly, we can prove ξ_L ((G, E) , (P, E)) ≤ ξ_L ((F, E) , (P, E)).

This completes the proof of Proposition 3.2.

Proposition 3.2 shows that ξ_L ((F, E) , (G, E)) is an example of ξ ((F, E) , (G, E)) and can be used for the proposed similarity measure S_ξ ((F, E) , (G, E)). Because ξ_L ((F, E) , (G, E)) is determined by φ ((F, E)) and ψ ((G, E)), Proposition 3.2 provides a convenient way to define ξ_L ((F, E) , (G, E)), as well as ξ ((F, E) , (G, E)), through φ ((F, E)) and ψ ((G, E)). In this paper, eight detailed expression examples for φ ((F, E)) and ψ ((G, E)) are given, as shown in Table 1, where $S_{ξ}^{i} (i = 1, 2, \dots, 8)$ denotes the difference names of the proposed similarity measure S_ξ ((F, E) , (G, E)) with ξ ((F, E) , (G, E)) = ξ_L ((F, E) , (G, E)).

Table 1

Examples of S_ξ with some kind of φ and ψ

S _ξ	φ ((F, E))	ψ ((G, E))
$S_{ξ}^{1}$	t_{F(e_i)} (u_j)	t_{G(e_i)} (u_j)
$S_{ξ}^{2}$	1 - f_{F(e_i)} (u_j)	1 - f_{G(e_i)} (u_j)
$S_{ξ}^{3}$	t_{F(e_i)} (u_j) (1 - f_{F(e_i)} (u_j))	t_{G(e_i)} (u_j) (1 - f_{G(e_i)} (u_j))
$S_{ξ}^{4}$	$\begin{matrix} m_{F (e_{i})} (u_{j}) = \\ \frac{t_{F (e_{i})} (u_{j}) + 1 - f_{F (e_{i})} (u_{j})}{2} \end{matrix}$	$\begin{matrix} m_{G (e_{i})} (u_{j}) = \\ \frac{t_{G (e_{i})} (u_{j}) + 1 - f_{G (e_{i})} (u_{j})}{2} \end{matrix}$
$S_{ξ}^{5}$	$\frac{t_{F (e_{i})} (u_{j}) + m_{F (e_{i})} (u_{j})}{2}$	$\frac{t_{G (e_{i})} (u_{j}) + m_{G (e_{i})} (u_{j})}{2}$
$S_{ξ}^{6}$	$\frac{1 - f_{F (e_{i})} (u_{j}) + m_{F (e_{i})} (u_{j})}{2}$	$\frac{1 - f_{G (e_{i})} (u_{j}) + m_{G (e_{i})} (u_{j})}{2}$
$S_{ξ}^{7}$	t_{F(e_i)} (u_j) m_{F(e_i)} (u_j)	t_{G(e_i)} (u_j) m_{G(e_i)} (u_j)
$S_{ξ}^{8}$	(1 - f_{F(e_i)} (u_j)) m_{F(e_i)} (u_j)	(1 - f_{G(e_i)} (u_j)) m_{G(e_i)} (u_j)

From Definition 2.5, we can easily prove that the variables φ ((F, E)) and ψ ((G, E)) presented in Table 1 satisfy the properties (C1), (C2) and (C3) given in Proposition 3.2.

3.2 Landmark preference based on similarity measure of vague soft sets

Landmark preference problems involve a set of landmarks denoted by G = {G₁, G₂, …, G_p}. Each landmark is measured by means of m feature properties, denoted by a parameter set E = {e₁, e₂, …, e_m} in VSSs. Let U = {u₁, u₂, …, u_n} be the universal set of elements, then the VSSs of G in the universe U are represented as follows: $(G_{j}, E) = {G_{j} (e_{i}) | i = 1, 2, \dots, m; j = 1, 2, \dots, p},$

where $\begin{matrix} G_{j} (e_{i}) = (\frac{[t_{G_{j} (e_{i})} (u_{1}), 1 - f_{G_{j} (e_{i})} (u_{1})]}{u_{1}}, \\ \frac{[t_{G_{j} (e_{i})} (u_{2}), 1 - f_{G_{j} (e_{i})} (u_{2})]}{u_{2}}, \\ . . ., \\ \frac{[t_{G_{j} (e_{i})} (u_{n}), 1 - f_{G_{j} (e_{i})} (u_{n})]}{u_{n}}) . \end{matrix}$

Accordingly, let (I, E) be the set of ideal solutions corresponding to (G_j, E) in the universe U, then the VSSs of (I, E) are represented as follows: $(I, E) = {I_{k} (e_{i}) | k = 1, 2, \dots, n; i = 1, 2, . \dots, m},$ where I _k(e_i) are the ideal solutions under the parameter e_i in the universe u _k.

The main computational procedures for landmark preference are as follows:

Step 1. Convert all property values into vague values, and construct VSSs for landmark preference.

Let r_ji (j = 1, 2, …, p i = 1, 2, …, m) represent all property values of the alternative landmarks and μ _ji represent the normalization of r_ji. If the benefit criteria are considered, the following equation [35] can be used to convert r _ji into μ _ji as follows: $μ_{ji} = (r_{ji} / r_{j max})^{2} .$ (2)

If the cost criteria are considered, the following equation [35] can be used to convert r _ji into μ _ji as follows: $μ_{ji} = {\begin{matrix} 1 - (r_{ji} / r_{j max})^{2}, r_{j min} = 0 \\ (r_{j min} / r_{ji})^{2}, r_{j min} \neq 0 . \end{matrix}$ (3)

Then, the following equation [36] can be used to convert μ _ji into vague values as follows: $t_{G_{j} (e_{i})} = (μ_{ji})^{2}, 1 - f_{G_{j} (e_{i})} = (μ_{ji})^{1 / 2} .$ (4)

The constructed VSSs of all the alternative landmarks are represented by (G_j, E).

Step 2. Determine the ideal solutions.

I _k(e_i) are determined as follows:

$\begin{matrix} I_{k} (e_{i}) = ([t_{G_{1} (e_{i})} (u_{k}), 1 - f_{G_{1} (e_{i})} (u_{k})] \lor (\land) [t_{G_{2} (e_{i})} (u_{k}), \\ 1 - f_{G_{2} (e_{i})} (u_{k})] \lor (\land) \dots \lor (\land) [t_{G_{p} (e_{i})} (u_{k}), \\ 1 - f_{G_{p} (e_{i})} (u_{k})]), \end{matrix}$ (5)

where the operators “∨” and “∧” depend on the criteria of landmark preference.

Step 3. Calculate the similarity measures S_j between (G_j, E) and (I, E).

Step 4. Rank all alternatives and determine the optimal landmark according to S_j.

4 Application to landmark preference

A real-world landmark preference example is adopted from [33], which includes seven alternative landmarks and eight kind of feature property values for each landmark, as shown in Table 2. The seven landmarks are denoted by Graben 29A, Graben 30, Graben 8, Graben 10, Graben 11, Graben 12 and Graben 29. The feature properties are denoted by α, β₁, β₂, γ_rgb, γ_hsb, δ, ɛ and ζ, where α denotes the facade size, β₁ is the shape factor, β₂ is the shape deviation, γ_rgb is the RGB color, γ_hsb is the HSB color, δ is the visibility, ɛ is the cultural importance and ζ is the identifiability. In this context, the proposed similarity measure of vague soft sets is used to choose the optimal landmark according to the feature properties, as described in Section 3.2.

Table 2
Property values of the landmarks

No. Landmark α β ₁ β ₂ γ _rgb γ _hsb δ ɛ ζ

1 Graben 29A 0.87 0.69 0 1.58 2.02 0 0 0.67

2 Graben 30 0.09 0.12 0 0 0.3 0 0 0

3 Graben 8 0.43 0.67 0 0.97 0 0.7 0.9 0

4 Graben 10 1.18 1.07 0 0.71 2.29 0 2.47 0.67

5 Graben 11 1.43 0.52 0 0.49 0.62 0.87 0 0.67

6 Graben 12 0.67 1.02 0 0.57 0.7 0.67 0 0.67

7 Graben 29 0 0 0 0.26 0.15 0 0 0

No.	Landmark	α	β ₁	γ _rgb	γ _hsb	δ	ɛ	ζ
1	Graben 29A	0.87	0.69	1.58	2.02	0	0	0.67
2	Graben 30	0.09	0.12	0	0.3	0	0	0
3	Graben 8	0.43	0.67	0.97	0	0.7	0.9	0
4	Graben 10	1.18	1.07	0.71	2.29	0	2.47	0.67
5	Graben 11	1.43	0.52	0.49	0.62	0.87	0	0.67
6	Graben 12	0.67	1.02	0.57	0.7	0.67	0	0.67
7	Graben 29	0	0	0.26	0.15	0	0	0

4.1 Procedures

First, let r_ji (j = 1, 2, …, 7 i = 1, 2, …, 8) represent the property values of the seven landmarks in the columns 3–10 of Table 2. Equations (2-4) are used to convert r_ji into vague values. The converted results of the seven landmarks are represented by (G_j, E) (j = 1, 2, …, 7). As an example, the converted results of landmark Graben 29A are shown in columns 3–10 of Table 3, where the vague values of the column e₃ (β₂) are set to vague value [0, 0] directly because the property values of β₂ are equal to 0 for all the seven landmarks. In Table 3, landmark Graben 29A and its properties of features are represented by a VSS (G₁, E) with the parameter set E = {e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈} = {a, β₁, β₂, γ_rgb, γ_hsb, δ, ɛ, ζ} in the universe U = {u₁, u₂} = {Benefit, Cost}.

Table 3
Property values of landmark Graben 29A represented by a VSS

VSS U e₁ (α) e₂ (β₁) e₃ (β₂) e₄ (γ_rgb) e₅ (γ_hsb) e₆ (δ) e₇ (ɛ) e₈ (ζ)

(G₁, E) u ₁ [0.1370, 0.6084] [0.1729, 0.6449] [0, 0] [1, 1] [0.6054, 0.8821] [0, 0] [0, 0] [1, 1]

u ₂ [0.3967, 0.7936] [0.3412, 0.7643] [0, 0] [0, 0] [0.0492, 0.4711] [1, 1] [1, 1] [0, 0]

VSS	U	e₁ (α)	e₂ (β₁)	e₃ (β₂)	e₄ (γ_rgb)	e₅ (γ_hsb)	e₆ (δ)	e₇ (ɛ)	e₈ (ζ)
(G₁, E)	u ₁	[0.1370, 0.6084]	[0.1729, 0.6449]	[0, 0]	[1, 1]	[0.6054, 0.8821]	[0, 0]	[0, 0]	[1, 1]
u ₂	[0.3967, 0.7936]	[0.3412, 0.7643]	[0, 0]	[0, 0]	[0.0492, 0.4711]	[1, 1]	[1, 1]	[0, 0]

Second, according to Equation (5), the ideal solutions are determined by adopting the operator “∨” over the universe u₁ and by adopting the operator “∧” over the universe u₂, respectively, as shown in Table 4.

Table 4

Ideal solution represented by a VSS

VSS	U	e₁ (α)	e₂ (β₁)	e₃ (β₂)	e₄ (γ_rgb)	e₅ (γ_hsb)	e₆ (δ)	e₇ (ɛ)	e₈ (ζ)
(I, E)	u ₁	[1, 1]	[1, 1]	[0, 0]	[1, 1]	[1, 1]	[1, 1]	[1, 1]	[1, 1]
u ₂	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0, 0]	[0, 0]

Then, the degrees of similarity between (G_j, E) and (I, E) are calculated by adopting the proposed similarity measures $S_{ξ}^{i} (i = 1, 2, \dots, 8)$ , respectively, as shown in Table 5.

Table 5

A comparison of the landmark preference results

Similarity measure	Graben 29A	Graben 30	Graben 8	Graben 10	Graben 11	Graben 12	Graben 29
S_C	0.5698	0.1372	0.3419	0.7369	0.5653	0.5012	0.1346
S_HK	0.5698	0.1372	0.3419	0.7369	0.5653	0.5012	0.1346
S_LX	0.5698	0.1372	0.3419	0.7369	0.5653	0.5012	0.1346
S_LO	0.5530	0.1369	0.3226	0.7273	0.5608	0.4846	0.1344
S_LC	0.5698	0.1372	0.3419	0.7369	0.5653	0.5012	0.1346
S_M	0.5698	0.1372	0.3419	0.7369	0.5653	0.5012	0.1346
S_LS1	0.5698	0.1372	0.3419	0.7369	0.5653	0.5012	0.1346
S_HY1	0.4929	0.1261	0.2308	0.6904	0.5177	0.4089	0.1260
S_Y	0.6109	0.1261	0.3602	0.7593	0.5277	0.5131	0.1261
S_BA	0.5698	0.1372	0.3419	0.7369	0.5653	0.5012	0.1346
S_CC	0.5927	0.1397	0.3789	0.7512	0.5853	0.5341	0.1369
S_SW	0.6069	0.1692	0.4172	0.7636	0.6163	0.5672	0.1568
S_F	0.8251	0.9772	0.7499	0.8959	0.8967	0.7966	0.9823
S_WQ	0.5698	0.1372	0.3419	0.7369	0.5653	0.5012	0.1346
S_MS	0.5629	0.1302	0.3308	0.7315	0.5576	0.4862	0.1298
S_MSSK	0.2518	0.0109	0.1232	0.3492	0.2549	0.2182	0.0082
$S_{ξ}^{1}$	0.5585	0.1344	0.3223	0.7298	0.5546	0.4856	0.1325
$S_{ξ}^{2}$	0.5572	0.1344	0.3206	0.7291	0.5542	0.4845	0.1325
$S_{ξ}^{3}$	0.5587	0.1344	0.3226	0.7299	0.5547	0.4858	0.1325
$S_{ξ}^{4}$	0.5578	0.1344	0.3214	0.7294	0.5544	0.4851	0.1325
$S_{ξ}^{5}$	0.5582	0.1344	0.3219	0.7296	0.5545	0.4854	0.1325
$S_{ξ}^{6}$	0.5575	0.1344	0.3210	0.7292	0.5543	0.4848	0.1325
$S_{ξ}^{7}$	0.5589	0.1344	0.3229	0.7300	0.5548	0.4860	0.1325
$S_{ξ}^{8}$	0.5583	0.1344	0.3220	0.7296	0.5545	0.4854	0.1325

Finally, the final order of the seven landmarks is ranked by the descending order of similarity measures, and the optimal landmark is obtained based on the principle of maximum of similarity measures. From Table 5, it is obvious that the order of the seven landmarks obtained by the proposed similarity measures $S_{ξ}^{i} (i = 1, 2, \dots, 8)$ is Graben 10, Graben 29A, Graben 11, Graben 12, Graben 8, Graben 30 and Graben 29, where the optimal landmark is landmark Graben 10. These results of landmark preference are the same as the ones obtained by Nothegger’s specialized constructed computational model presented in [33].

4.2 Comparative analysis

To obtain a comparison of the landmark preference results, the existing similarity measures in [16–24 , 38] are also used to make a calculation, as shown in Table 5, where the existing similarity measures are denote by S_C, S_HK [38], S_LX [27], S_LO [31], S_LC [21], S_M [17], S_LS₁ [30], S_HY₁ [16], S_Y [18], S_BA [19], S_CC [20], S_SW [22], S_F [23], S_WQ [26], S_MS [24] and S_MSSK [28], respectively. The parameters and weights of the set of similarity measures S_LC [21], S_M [17], S_LS₁ [30], S_BA [19] and S_CC [20] are as follows: p = 1 in S_LC, S_M and S_LS₁; p = 1 and t = 2 in S_BA; w₁= w₂ = w₃ = 1/3 in S_CC. The readers can refer to [16–24 , 38] for more details about these existing similarity measures.

From Table 5, we can observe that the similarity measures S_C [37], S_HK [38], S_LX [27], S_LC [21], S_M [17], S_LS₁ [30], S_BA [19], S_CC [20], S_WQ [26] and S_MS [24], have the same conclusions as the proposed method’s results. However, the similarity measures S_C [37], S_HK [38], S_LX [27], S_LC [21], S_M [17], S_LS₁ [30], S_BA [19] and S_WQ [26], have same degrees of similarity between (G_j, E) and (I, E), respectively, indicating there is no difference between them and it does not matter what term we choose in this example. In contrast, all the proposed similarity measures $S_{ξ}^{i} (i = 1, 2, \dots, 8)$ can make a difference in term of degrees of similarity for each other. Besides, the similarity measures S_LO [31], S_HY₁ [16], S_SW [22] and S_MSSK [28] are unreasonable cases, where the order between landmark Graben 29A and landmark Graben 11 is reversed. The similarity measure S_Y [18] is unable to differentiate the order between landmark Graben 30 and landmark Graben 29. It is noticeable that the order of the seven alternative landmarks obtained by the similarity measure S_F [23] is completely unreasonable. The unreasonable cases are highlighted in bold type in Table 5.

The landmark preference above reflects the overall performance of the similarity measures when measuring similarity between two VSSs. It is a comprehensive result based on multi-element sets. An effective approach to find the problems based on multi-element sets is to analyze each problem based on single-element sets, because a comparison based on single-element sets is the basis for multi-element sets, it and can directly demonstrate drawbacks [31]. In [23], Nguyen found that the similarity measures S_C [37], S_HK [38], S_LX [27], S_LO [31], S_HY₁ [16], S_CC [20] and those similarity measures with one or two parameters, i.e., S_LC [21], S_M [17], S_LS₁ [30] and S_BA [19], do not differentiate between certain pairs of different singleton vague sets, i.e., for the two pairs where A₁ = {(u, [0.3, 0.7])} , B₁ = {(u, [0.4, 0.6])} and A₂ = {(u, [0.4, 0.8])} , B₂ = {(u, [0.5, 0.7])} in U = {u}. Nguyen [23] also found that the similarity measure S_Y [18] has a problem with “the division by zero” for the pair where A = {(u, [0.5, 0.5])} , B = {(u, [0, 1])}.

In this paper, the drawbacks of the similarity measures S_SW [22], S_F [23], S_WQ [26], S_MS [24] and S_MSSK [28] have been determined. Let us consider the cases where A₁ = {(u, [0.2, 0.4])} , B₁ = {(u, [0.1, 0.3])} and A₂ = {(u, [0.2, 0.4])} , B₂ = {(u, [0.2, 0.3])} in U = {u}, the results for the similarity measure S_SW [22] can be obtained as follows: S_SW (A₁, B₁) = S_SW (A₂, B₂) =0.992, indicating it does not differentiate these pairs. Consider the cases where A₁ = {(u, [0.4, 0.7])} , B₁ = {(u, [0.6, 0.9])} and A₂ = {(u, [0.2, 0.7])} , B₂ = {(u, [0.2, 0.6])} in U = {u}, the results for the similarity measure S_F [23] can be obtained as follows: S_F (A₁, B₁) =0.9525, S_F (A₂, B₂) =0.9067 and S_F (A₁, B₁) > S_F (A₂, B₂). It is obviously an unreasonable case because A₂ should be more similar to B₂, and this degree of similarity should be larger than the one between A₁ and B₁. Similarly, consider the cases where A₁ = {(u, [0.4, 0.8])} , B₁ = {(u, [0.3, 0.9])} and A₂ = {(u, [0.4, 0.8])} , B₂ = {(u, [0.5, 0.7])} in U = {u}, the results for the similarity measure S_MS [24] can be obtained as follows: S_MS (A₁, B₁) = S_MS (A₂, B₂) =0.9455, indicating it does not differentiate these pairs. Consider the cases where A₁ = {(u, [0.4, 0.7])} , B₁ = {(u, [0.6, 0.9])} and A₂ = {(u, [0.4, 0.9])} , B₂ = {(u, [0.6, 0.7])} in U = {u}, the results of the similarity measure S_MSSK [28] can be obtained as follows: S_MSSK (A₁, B₁) = S_MSSK (A₂, B₂) =0.7436, indicating it does not differentiate these pairs. Additionally, the similarity measure S_WQ [26] has the same unreasonable case as the similarity measure S_LX [27] shown in [23] because the former is based on the latter. In contrast, as shown in Table 6, the proposed similarity measures $S_{ξ}^{i} (i = 1, 2, \dots, 8)$ can differentiate the pairs mentioned above and in [23].

Table 6
Similarity measure between A and B

pair 1 2 3 4 5 6 7 8 9 10 11

A [0.3,0.7] [0.5,0.5] [0.4,0.8] [0,0.13] [0.6,0.73] [0.2,0.4] [0.2,0.4] [0.4,0.7] [0.2,0.7] [0.4,0.8] [0.4,0.9]

B [0.4,0.6] [0,1] [0.5,0.7] [0.28,0.45] [0.28,0.45] [0.1,0.3] [0.2,0.3] [0.6,0.9] [0.2,0.6] [0.3,0.9] [0.6,0.7]

$S_{ξ}^{1}$ 0.9447 0.7031 0.9444 0.6923 0.6904 0.8970 0.9477 0.7880 0.9433 0.9413 0.8880

$S_{ξ}^{2}$ 0.9436 0.6875 0.9432 0.6913 0.6896 0.8965 0.9474 0.7857 0.9404 0.9381 0.8857

$S_{ξ}^{3}$ 0.9454 0.7109 0.9449 0.6932 0.6914 0.8973 0.9480 0.7884 0.9436 0.9417 0.8893

$S_{ξ}^{4}$ 0.9444 0.7031 0.9440 0.6918 0.6900 0.8967 0.9476 0.7869 0.9418 0.9400 0.8876

$S_{ξ}^{5}$ 0.9446 0.7031 0.9442 0.6921 0.6902 0.8969 0.9477 0.7874 0.9425 0.9406 0.8878

$S_{ξ}^{6}$ 0.9440 0.6953 0.9436 0.6916 0.6898 0.8966 0.9475 0.7863 0.9411 0.9391 0.8867

$S_{ξ}^{7}$ 0.9455 0.7109 0.9451 0.6934 0.6916 0.8973 0.9480 0.7891 0.9439 0.9423 0.8896

$S_{ξ}^{8}$ 0.9449 0.7031 0.9444 0.6930 0.6911 0.8972 0.9479 0.7874 0.9426 0.9404 0.8881

pair	1	2	3	4	5	6	7	8	9	10	11
A	[0.3,0.7]	[0.5,0.5]	[0.4,0.8]	[0,0.13]	[0.6,0.73]	[0.2,0.4]	[0.2,0.4]	[0.4,0.7]	[0.2,0.7]	[0.4,0.8]	[0.4,0.9]
B	[0.4,0.6]	[0,1]	[0.5,0.7]	[0.28,0.45]	[0.28,0.45]	[0.1,0.3]	[0.2,0.3]	[0.6,0.9]	[0.2,0.6]	[0.3,0.9]	[0.6,0.7]
$S_{ξ}^{1}$	0.9447	0.7031	0.9444	0.6923	0.6904	0.8970	0.9477	0.7880	0.9433	0.9413	0.8880
$S_{ξ}^{2}$	0.9436	0.6875	0.9432	0.6913	0.6896	0.8965	0.9474	0.7857	0.9404	0.9381	0.8857
$S_{ξ}^{3}$	0.9454	0.7109	0.9449	0.6932	0.6914	0.8973	0.9480	0.7884	0.9436	0.9417	0.8893
$S_{ξ}^{4}$	0.9444	0.7031	0.9440	0.6918	0.6900	0.8967	0.9476	0.7869	0.9418	0.9400	0.8876
$S_{ξ}^{5}$	0.9446	0.7031	0.9442	0.6921	0.6902	0.8969	0.9477	0.7874	0.9425	0.9406	0.8878
$S_{ξ}^{6}$	0.9440	0.6953	0.9436	0.6916	0.6898	0.8966	0.9475	0.7863	0.9411	0.9391	0.8867
$S_{ξ}^{7}$	0.9455	0.7109	0.9451	0.6934	0.6916	0.8973	0.9480	0.7891	0.9439	0.9423	0.8896
$S_{ξ}^{8}$	0.9449	0.7031	0.9444	0.6930	0.6911	0.8972	0.9479	0.7874	0.9426	0.9404	0.8881

The drawbacks existing in single-element set tests for the existing similarity measures lead them to be less reliable and less accurate in terms of the overall performance of the landmark preference, except for the similarity measures S_CC and S_MS [24]. However, let us consider the following pattern recognition problem adopted from [23]. Let U = {u₁, u₂, u₃} be the universal set and E be the set of parameters given by E = {e1}. Three given patterns, represented by VSSs in the universe U, are $F_{1} (e_{1}) = (\frac{[1.0, 1.0]}{u_{1}}, \frac{[0.8, 1.0]}{u_{2}}, \frac{[0.7, 0.9]}{u_{3}}),$ $F_{2} (e_{1}) = (\frac{[0.35, 0.85]}{u_{1}}, \frac{[0.45, 0.75]}{u_{2}}, \frac{[0.55, 0.65]}{u_{3}}),$ $F_{3} (e_{1}) = (\frac{[0.25, 0.75]}{u_{1}}, \frac{[0.35, 0.65]}{u_{2}}, \frac{[0.45, 0.55]}{u_{3}}) .$

The sample to be recognized is $G (e_{1}) = (\frac{[0.3, 0.8]}{u_{1}}, \frac{[0.4, 0.7]}{u_{2}}, \frac{[0.5, 0.6]}{u_{3}}) .$

Our aim is to classify the sample G(e₁) into one of the known different patterns F_j (e₁) (j = 1, 2, 3). The results can be obtained as follows: S_CC (F₁, G) =0.6408, S_CC (F₂, G) = S_CC (F₃, G) =0.9650 and S_MS (F₁, G) =0.6588, S_MS (F₂, G) = S_MS (F₃, G) =0.95, indicating the similarity measures S_CC [20] and S_MS [24] do not classify the sample G(e₁) into one of the known patterns because they have equally calculated maximal similarity measures. The drawbacks existing in the similarity measures S_CC [20] and S_MS [24] may lead to some unreasonable cases that apply to other real-world applications. In contrast, despite subtle difference in degrees of similarity between $S_{ξ}^{i} (F_{2}, G)$ and $S_{ξ}^{i} (F_{3}, G) (i = 1, 2, \dots, 8),$ i.e., $S_{ξ}^{1} (F_{1}, G) = 0 . 6247$ , $S_{ξ}^{1} (F_{2}, G) = 0.9474$ and $S_{ξ}^{1} (F_{3}, G) = 0.9475$ , the proposed similarity measures $S_{ξ}^{i} (i = 1, 2, \dots, 8)$ can classify the sample G(e₁) into the pattern F₃(e₁), which is the same as the one obtained by the similarity measure S_F presented in [23].

Based on the above comparison analysis, we observed that the proposed similarity measures can be successfully utilized in landmark preference, and thus can avoid constructing and using a specialized computational model to address the landmark preference problems as in [33]. By using the subitem ξ_L ((F, E) , (G, E)), the proposed similarity measures $S_{ξ}^{i} (i = 1, 2, \dots, 8)$ are more reliable and more accurate in dealing with landmark preference problems than other existing similarity measures. Moreover, the characteristic that can derive different detailed expressions makes the proposed similarity measure S_ξ ((F, E) , (G, E)) more flexible and effective in measuring similarity among single-element set tests, pattern recognition problems and landmark preference compared to those of the existing similarity measures.

5 Discussion

Although the proposed similarity measures $S_{ξ}^{i} (i = 1, 2, \dots, 8)$ show that they all can differentiate the pairs in Table 6, it should be noticed that this is not always the case. For example, consider the cases where A₁ = {(u, [0.6, 0.8])} , B₁ = {(u, [0.5, 0.8])} and A₂ = {(u, [0.6, 0.8])} , B₂ = {(u, [0.6, 0.9])} in U = {u}, the results of $S_{ξ}^{1}$ and $S_{ξ}^{5}$ show that they do not differentiate these two pairs, where $S_{ξ}^{1} (A_{1}, B_{1}) = S_{ξ}^{1} (A_{2}, B_{2}) = 0.9450$ and $S_{ξ}^{5} (A_{1}, B_{1}) = S_{ξ}^{5} (A_{2}, B_{2}) = 0.9448$ . However, from another point of view, it just demonstrates the advantages of the proposed similarity measure S_ξ ((F, E) , (G, E)) that can derive different detailed expressions by using the subitem ξ_L ((F, E) , (G, E)), as well as ξ ((F, E) , (G, E)), indicating aside from $S_{ξ}^{1}$ and $S_{ξ}^{5}$ , the rest of $S_{ξ}^{i}$ can be used to calculate the similarity measures and differentiate these two pairs.

Clearly, aside from $S_{ξ}^{i} (i = 1, 2, \dots, 8)$ , other forms of detailed expressions can be derived from the subitem ξ_L ((F, E) , (G, E)) only if they satisfy the three properties (R1), (R2) and (R3) defined in Definition 3.1. For example, for any (F, E) , (G, E) , (P, E) ∈ VSS (U) , if (F, E) ⊆ (G, E) ⊆ (P, E), another similar detailed expressions can be derived by replacing max(φ ((F, E)) , ψ ((G, E))) with 1 - min (φ ((F, E)) , ψ ((G, E))) in ξ_L ((F, E) , (G, E)), due to the relations 1 - min(t_{F(e_i)} (u_j) , t_{G(e_i)} (u_j)) =1 - t_{F(e_i)} (u_j) =1 - min(t_{F(e_i)} (u_j) , t_{P(e_i)} (u_j)) and 1 - min(t_{G(e_i)} (u_j) , t_{P(e_i)} (u_j)) = 1 - t_{G(e_i)} (u_j) ≤1 - min (t_{F(e_i)} (u_j) , t_{P(e_i)} (u_j)) =1 - t_{F(e_i)} (u_j). Although these derived detailed expressions may not be able to overcome the drawbacks of having unreasonable cases completely, some real-world applications can benefit from the use of this type of variable subitem, as presented in the landmark preference example.

6 Conclusion

This work presented a new type of similarity measure between two vague soft sets and its properties. Compared with the existing similarity measures, the proposed similarity measure contains the subitem satisfying certain properties rather than the single value parameters defined within a range of values. In this way, some different detailed expressions, including but not limited to the detailed expressions given in this paper, can be derived from the subitem. The advantage of this change lies in the different equations for calculating similarity measure, which can be constructed correspondingly by choosing different detailed expressions for the subitem according to different practical applications. For example, based on the characteristic of similarity measures $S_{ξ}^{1}$ and $S_{ξ}^{2}$ , the similarity measure $S_{ξ}^{1}$ can be chosen for those true membership first applications. In contrast, the similarity measure $S_{ξ}^{2}$ can be chosen for those false membership first applications. As a real-world application, the proposed similarity measure is utilized to address the landmark preference problems in a vague soft set environment. This significance lies in this technique of similarity measure on vague soft sets being useful and effective for handling realistic problems, as is the one using the specialized computational model. The comparisons and analysis of the landmark preference example provide advantages of the proposed similarity measure over the existing similarity measures.

Aside from the feature properties of a landmark presented in Table 2, some other feature properties can be represented by linguistic variables or qualitative values rather than numerical values. Therefore, future research will consider combining similarity measure of vague soft sets with qualitative tools to deal with landmark preference problems.

Conflict of interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgments

The works described in this paper are supported by the National Natural Science Foundation of China under Grant no. 41461085; the Natural Science Foundation of Guangxi Province under Grant no. 2016GXNSFAA380035; the Foundation of Guangxi Key Laboratory of Spatial Information and Geomatics under Grant no. 16-380-25-04; the “BaGui Scholars” Special Funds of Guangxi Province under Grant no. 2013-3; the Doctoral Foundation of Guilin University of Technology under Grant no.1996015.

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