Entropy measures for hesitant fuzzy sets and their application in multi-criteria decision-making

Abstract

Entropy is used to measure the uncertain degree of fuzzy sets and has been widely used in many fields. This paper introduces an axiomatic definition of entropy measure and a novel entropy formula for hesitant fuzzy elements (HFEs). Afterwards, a general form of entropy measures for HFEs is proposed, from which a family of concrete entropy formulas for HFEs can be derived. Compared with the existing ones, these formulas can measure both fuzziness and hesitation of HFEs, as a result, the uncertain information can be described in a more appropriate manner. The proposed axiomatic definition and entropy formulas are used to define the entropy measure for hesitant fuzzy sets. A multi-criteria decision making model which uses the proposed entropy measures for HFEs to compute the criteria weights and obtain a ranking of alternatives is introduced.

Keywords

Hesitant fuzzy set entropy measure multi-criteria decision-making

1 Introduction

Since Zadeh [35] introduced fuzzy sets, many generalized forms have been proposed, among which there are interval-valued fuzzy sets (IVFSs) [3], intuitionistic fuzzy sets (IFSs) [1], interval-value intuitionistic fuzzy sets (IVIFS) [2], vague sets [10], type-2 fuzzy sets [17], type-n fuzzy sets [7] and fuzzy multisets [18]. All these extensions are based on the same rationale that defining a fuzzy set is not an easy task because there is no a clear way to assign the membership degree of an element to a fixed set [23, 24].

Recently, a new extension of fuzzy sets, so-called Hesitant Fuzzy Set (HFS), has been proposed in [23, 24]. The motivation is that when defining the membership of an element, the difficulty of establishing the membership degree is not because there is a margin of error, or some possibility distribution on the possible values, but because there is a set of possible values [23]. HFSs can be used when experts hesitate among several values to provide their preferences over the alternatives and the use of only one value is not enough to reflect their knowledge in a suitable way.

In the last years, HFSs have attracted much attention of researchers and different proposals have been introduced in the literature such as, distance measures, similarity measures, and entropy measures [14 , 32], different aggregation operators [12 , 29], decision making methods that deal with HFS [4 , 26–28], even it has been extended to deal with linguistic information [15 , 20].

This paper is focused on the entropy measure for HFSs that is a very important notion for measuring uncertain information, along with a lot of research productions of entropy measures for FSs and IFSs. Zadeh first used fuzziness in an entropy measure which was mentioned in [35]. Then, De Luca and Termini [6] proposed the axioms that the fuzzy entropy should comply and they defined the entropy of a fuzzy set based on Shannon’s function. Yager [33] presented an entropy measure to view the fuzziness degree of the fuzzy set in terms of a lack of distinction between the fuzzy set and its complement. Other entropies for fuzzy sets with different points of views can be found in[8, 19, 22].

In hesitant fuzzy environment, Xu and Xia [32] proposed an entropy axiomatic definition and some formulas for hesitant fuzzy elements (HFEs). Farhadinia [9] presented some counterexamples to point out that the entropies proposed in [32] cannot discriminate some HFEs, even though they are apparently different and proposed a series of entropy measures for HFSs, which are based on the distance measures for HFSs. However, it can be shown that Farhadinia’s entropy measures fail to discriminate HFEs that have the same distance with the HFE {0.5}. Therefore, in order to overcome this shortcoming, this paper aims to propose new entropy axiomatic definitions and construct entropy formulas of HFEs and HFSs, which can reflect the uncertainty of HFEs and HFSs in a proper way. Based on these entropy formulas, an approach to solve multi-criteria decision-making problems with unknown criteria weights is introduced.

HFS is a simple and effective tool used to express experts’ hesitation in decision-making. Suppose two experts discuss the membership degree of an alternative for a criterion, and one of them wants to assign 0.1 and the other 0.8. Then the membership degree of the alternative for the criterion might be described by the HFE {0.1, 0.8}, which describes the uncertain information of experts’ preference. Therefore, it is necessary to depict the uncertainty degree associated to the HFE {0.1, 0.8} in an appropriate way. Since HFS is a generalized form of FS, the fuzziness as a characteristic of FS, is also an important index for HFS. Besides, HFE has its own characteristic, that is the hesitation among the possible membership degrees 0.1 and 0.8. Thus, in order to depict the uncertainty reflected by an HFE, the fuzziness and hesitation of information should be considered. Therefore, the proposed axiomatic definitions and entropy measures can depict both fuzziness and hesitation of HFEsand HFSs.

The rest of this paper is organized as follows. Section 2 reviews some concepts of HFSs and distance measures of HFSs. Section 3 proposes a new axiomatic definition and an entropy measure for HFEs. Subsequently, a family of entropy measures based on the previous one is introduced. These proposals are generalized to define an axiomatic definition and entropy measures for HFSs. Section 4 compares the proposed axiomatic definition of entropy and concrete entropy formulas for HFEs with those defined in [9, 32]. Section 5 puts forward a method for multi-criteria decision-making which uses entropy measures for HFSs to obtain a ranking of alternatives. Section 6 introduces an illustrative example to demonstrate the practicality and effectiveness of the proposed formulas in a multi-criteria decision-making problem. Finally, this paper is concluded in Section 7.

2 Preliminaries

This section reviews some basic concepts, such as Hesitant Fuzzy Sets (HFSs) [23] and distance measures for HFSs [31]. Throughout this paper, X = {x₁, x₂, …, x_n} is used to denote the universe of discourse.

In [23] it was introduced a new extension of fuzzy set with the goal of modelling the uncertainty originated by the hesitation that might arise in the assignment of the membership degrees of the elements to a fuzzy set.

Definition 1. [23] Let X be a discourse set. A HFS on X is defined in terms of a function h that when is applied to X, it returns a non-empty subset of values in the interval [0, 1].

To be easily understood, Xia and Xu [30] expressed an HFS by $A = {〈 x_{i}, h_{A} (x_{i}) 〉 | x_{i} \in X},$ (1) where h_A (x_i) is a set of some values in [0, 1], denoting the possible membership degrees of the element x_i ∈ X to the set A. Especially, if there is only one value in each h_A (x_i) (i = 1, 2, …, n), then the HFS A is reduced to a fuzzy set, which indicates that fuzzy sets are a special type of HFSs. HFS(X) is the set of all the HFSs on X.

For convenience, Xia and Xu [30] named h_A (x_i), abbreviated to h, a hesitant fuzzy element (HFE). For any HFE h = {h¹, h², …, h^{l
_h}}, we assume that hⁱ< h^j, for ∀i < j (i, j = 1, 2, …, l_h) in the whole paper, where l_h, abbreviated to l in the case of no confusion, is the number of values in h and is called the length of h. H is the set of all the HFEs

Example 1. Let X = {x₁, x₂, x₃} be the discourse set. Then A = {〈x₁, {0.2, 0.3} 〉, 〈x₂, {0.1, 0.2} 〉, 〈x₃, {0.6, 0.7, 0.8} 〉} is an HFS on X.

Let A = {〈x_j, h_A (x_j) 〉|x_j ∈ X} and B = {〈x_j, h_B (x_j) 〉|x_j ∈ X} be two HFSs. In general, the lengths of h_A (x_j) and h_B (x_j) may be different. Let l_{x
_j} = max {l_{h_A(x_j)}, l_{h_B(x_j)}}. In order to operate correctly, h_A (x_j) and h_B (x_j) should have the same length l_{x
_j}. To do this, the shorter one is extended until the lengths of both are the same. The best way to extend the shorter one is to add the same element in it until the changed HFE has the same length as the longer one. Any value in the shorter one might be added to extend it. In this paper, it is considered an optimistic point of view in which the shorter one is extended by repeating its maximum element.

For two HFSs A and B, we assume that h_A (x_j) and h_B (x_j) have the same length l_{x
_j}. Let $h_{A}^{i} (x_{j})$ and $h_{B}^{i} (x_{j})$ be the ith smallest values in h_A (x_j) and h_B (x_j), respectively. Xu and Xia [31] gave a variety of distance measures for HFSs, some of which are described as follows:

The generalized hesitant normalized distance: $d_{g h} = (A, B) = {\frac{1}{n} \sum_{j = 1}^{n} (\frac{1}{l_{x_{j}}} \sum_{i = 1}^{l_{x_{j}}} | h_{A}^{i} (x_{j}) - h_{B}^{i} (x_{j}) |^{λ})}^{\frac{1}{λ}}$ (2)

where λ > 0.

The generalized hesitant weighted distance: $d_{g h w} = (A, B) = {\sum_{j = 1}^{n} w_{j} (\frac{1}{l_{x_{j}}} \sum_{i = 1}^{l_{x_{j}}} | h_{A}^{i} (x_{j}) - h_{B}^{i} (x_{j}) |^{λ})}^{\frac{1}{λ}}$ (3)

where λ > 0, and ω_j (j = 1, 2, …, n) is the weight of the element x_j with ω_j ∈ [0, 1] and $\sum_{j = 1}^{n} ω_{j} = 1$ .

More information about distance measures for HFS can be found in [21, 31].

3 Entropy measure for HFSs

In order to investigate new entropy measures for HFSs, first some entropy measures for HFEs are studied and a new axiomatic definition of the entropy measure for HFEs is proposed. Furthermore, we put forward a family of entropy measures and a series of entropy formulas for HFEs. Finally, entropy measures for HFSs based on the entropy measures for HFEs are proposed. The efficiency of the proposed entropy measures is demonstrated through comparisons with some existing entropy measures in [9] and [31].

3.1 Entropy measures for HFEs

For an HFE h = {h¹, h², …, h^l}, its uncertainty of information should include two facts: the fuzziness and hesitation of information. The fuzziness is dominated by the difference between the averaging value of the elements in h and the most fuzzy value {0.5}. The hesitation is reflected by the deviation degree of the elements in h. The averaging value and the deviation function value of h are defined as follows.

Definition 2. [12] Let h = {h¹, h², …, h^l} be an HFE, then

the averaging value (score function value) of an HFE h is defined as $θ (h) = \frac{1}{l} \sum_{i = 1}^{l} h^{i} .$ (4)

the deviation function value of an HFE h is defined as $η (h) = \frac{2}{l (l - 1)} \sum_{i = 1}^{l - 1} \sum_{j = i + 1}^{l} (h^{j} - h^{i}) .$ (5)

Based on the score function θ (h) and the deviation function η (h), an entropy axiomatic definition for HFEs is proposed.

Definition 3. The entropy on an HFE h is a real-valued function E : H → [0, 1], satisfying the following axiomatic requirements:

E (h) =0, if and only if h = {0} or h = {1};

E (h) =1, if and only if θ (h) =0.5;

E (h₁) ≤ E (h₂), if θ (h₁) ≤ θ (h₂) and η (h₁) ≤ η (h₂) for θ (h₂) ≤0.5, or θ (h₁) ≥ θ (h₂) and η (h₁) ≤ η (h₂) for θ (h₂) ≥0.5;

E (h) = E (h^c), where h^c is the complement of an HFE h and is defined by h^c= ⋃ _γ∈h {1 - γ}.

Remark 1. The axiomatic requirement (E3) reflects that for any two HFEs h₁ and h₂, if θ (h₁) is further from 0.5 than θ (h₂), that is, |θ (h₁) -0.5| > |θ (h₂) -0.5|, and the deviation function value of h₁ is less than η (h₂), then the uncertainty of h₁ should be less than h₂.

The new entropy measure for HFEs is defined and its properties are studied.

Theorem 1. For each HFE h ∈ H, E (h) is defined by $E (h) = \frac{1 - | cos (θ (h) \cdot π) | + η (h)}{1 + η (h)} .$ (6)

Then E is an entropy measure for HFEs.

Proof. It is sufficient to show the mapping E (h), defined by Equation 6, for satisfying the conditions (E1)–(E4) in Definition 3.

Since $θ (h) = \frac{1}{l} \sum_{i = 1}^{l} h^{i}$ and $η (h) = \frac{2}{l (l - 1)} \sum_{i = 1}^{l - 1} \sum_{j = i + 1}^{l} (h^{j} - h^{i})$ for an HFE h = {h¹, h², …, h^l}, we have 0 ≤ θ (h) ≤1, 0 ≤ η (h) ≤1 . Hence, $0 \leq \frac{1 - | cos (θ (h) \cdot π) | + η (h)}{1 + η (h)} \leq 1 .$

$E (h) = \frac{1 - | cos (θ (h) \cdot π) | + η (h)}{1 + η (h)} = 0$ if and only if | cos(θ (h) · π) | - η (h) =1 . Since 0 ≤ η (h) ≤ 1 and 0 ≤ | cos(θ (h) · π) | ≤1, we have -1 ≤ | cos(θ (h) · π) | - η (h) ≤ 1 . Therefore, | cos(θ (h) · π) | - η (h) =1 if and only if η (h) =0 and | cos(θ (h) · π) |=1, i.e., h = {0} or h = {1}.

$E (h) = \frac{1 - | cos (θ (h) \cdot π) | + η (h)}{1 + η (h)} = 1$ if and only if cos(θ (h) · π) =0, that is, θ (h) =0.5 .

Let θ (h) = θ, η (h) = η, the Equation 6 can be noted as $E (h) = E (θ, η) = \frac{1 - | cos (θ π) | + η}{1 + η} .$

The partial derivative of E (θ, η) with respect to θ is as follows, $\frac{\partial E (θ, η)}{\partial θ} = {\begin{matrix} \frac{π sin (θ π) (1 + η)}{(1 + η)^{2}}, 0 < θ \leq 0.5, \\ \frac{- π sin (θ π) (1 + η)}{(1 + η)^{2}}, 0.5 \leq θ < 1 . \end{matrix}$

Since 0 < sin(θπ) ≤1 and 1 ≤ 1 + η ≤ 2 for 0 < θ < 1, we can get $\frac{\partial E (θ, η)}{\partial θ} > 0$ if 0 < θ ≤ 0.5; $\frac{\partial E (θ, η)}{\partial θ} < 0$ if 0.5 ≤ θ < 1. Therefore, E (θ, η) is strictly monotone increasing with respect to θ ∈ (0, 0.5], and strictly monotone decreasing with respect to θ ∈ [0.5, 1).

On the other hand, the partial derivative of E (θ, η) with respect to η ∈ [0, 1] is denoted as follows, $\frac{\partial E (θ, η)}{\partial η} = \frac{| cos (θ π) |}{(1 + η)^{2}} .$

Clearly, $\frac{\partial E (θ, η)}{\partial η} \geq 0$ and E (θ, η) is monotonically increasing with respect to η ∈ [0, 1].

From the above discussion, it is easy to get that if θ (h₁) ≤ θ (h₂) ≤ 0.5 and η (h₁) ≤ η (h₂), then E (θ (h₁), η (h₁)) ≤ E (θ (h₁), η (h₂)) ≤ E (θ (h₂), η (h₂)), that is, E (h₁) ≤ E (h₂); if θ (h₁) ≥ θ (h₂) ≥ 0.5 and η (h₁) ≤ η (h₂), then E (θ (h₁), η (h₁)) ≤ E (θ (h₁), eta (h₂)) ≤ E (θ (h₂), η (h₂)), that is, E (h₁) ≤ E (h₂).

For an HFE h = {h¹, h², …, h^l}, we have θ (h^c) = $\frac{1}{l} \sum_{i = 1}^{l} (1 - h^{i})$ = 1 - θ (h) and η (h^c) = (1 - h^l) - (1 - h¹) = η (h) . Therefore,

$\begin{matrix} E (h^{c}) & = & \frac{1 - | cos (θ (h^{c}) \cdot π) | + η (h^{c})}{1 + η (h^{c})} \\ = & \frac{1 - | cos ((1 - θ (h)) \cdot π) | + η (h)}{1 + η (h)} \\ = & \frac{1 - | - cos (θ (h) \cdot π) | + η (h)}{1 + η (h)} \\ = & \frac{1 - | cos (θ (h) \cdot π) | + η (h)}{1 + η (h)} = E (h) . \end{matrix}$

Equation 6 satisfies the conditions (E1)–(E4) in Definition 4, thus E is an entropy measure for HFEs.

Corollary 1. E (h) is strictly monotone increasing with respect to θ ∈ (0, 0.5] and strictly monotone decreasing with respect to θ ∈ [0.5, 1). In addition, it is monotonically increasing with respect to η ∈ [0, 1].

The conclusion is obviously shown in the proof process of Theorem 1.

From Corollary 1, we know that the greater the difference between θ (h) and 0.5 is, the greater E (h) is; the greater the deviation function value η (h) is, the greater E (h) is. Since the fuzziness of an HFE h is dominated by the difference between the score function value θ (h) and the most fuzzy value 0.5, and the hesitation is reflected by the deviation function value η (h), it is obvious that the entropy measure E (h) considers both the fuzziness and hesitation of anHFE h.

3.2 A family of entropy measures for HFEs

In this subsection, a family of entropy measures for HFEs based on the entropy measure E ispresented. Additionally, four concrete entropy measures are studied.

Theorem 2. For each HFE h, let

$E_{g} (h) = \frac{f (θ (h)) + k η (h)}{1 + k η (h)},$ (7)

where k ∈ [0, 1]. Then E_g (h) is an entropy measure for an HFE h, where the function f : [0, 1] → [0, 1] satisfies the following three conditions:

f (1 - x) = f (x);

f (x) is strictly monotone increasing with respect to x ∈ (0, 0.5] and strictly monotone decreasing with respect to x ∈ [0.5, 1);

It interpolates three points, (0, 0), $(\frac{1}{2}, 1)$ and (1, 0).

Corollary 2. E_g (h) is monotonically increasing with respect to θ ∈ (0, 0.5], and monotonically decreasing with respect to θ ∈ [0.5, 1). In addition, it is monotonically increasing with respect to η ∈ [0, 1].

It is noted that if we change the function f (x) in E_g (h) defined by Equation 7, we can obtain a series of entropy measures for HFEs. For instance, let k = 1 and let f (x) =1 - |1 - 2x|, $f (x) = 1 - 4 (x - \frac{1}{2})^{2}$ , f (x) =1 - | cos(πx) | and f (x) = sin(πx), respectively.

Four different entropy formulas are then obtained:

$E_{g 1} (h) = \frac{1 - | 1 - 2 θ (h) | + η (h)}{1 + η (h)},$ (8)

$E_{g 2} (h) = E (h) = \frac{1 - | cos (θ (h) \cdot π) | + η (h)}{1 + η (h)},$ (9)

$E_{g 3} (h) = E_{2} (h) = \frac{sin (θ (h) \cdot π) + η (h)}{1 + η (h)},$ (10)

$E_{g 4} (h) = \frac{1 - 4 (θ (h) - \frac{1}{2})^{2} + η (h)}{1 + η (h)} .$ (11)

Figure 1 shows the graphs of the four entropy formulas, E_g1 (h), E_g2 (h), E_g3 (h) and E_g4 (h). To do so, η (h) is a fixed value.

From Fig. 1, it is easy to see that E_g1 (h), E_g2 (h), E_g3 (h) and E_g4 (h) are all monotonically increasing with respect to θ ∈ (0, 0.5], and monotonically decreasing with respect to θ ∈ [0.5, 1). However, the increasing degree and decreasing degree of the four functions with respect to θ are different. When θ (h) closes to 0.5, the value of E_g3 (h) increases slowly for θ (h) ≤0.5 and decreases slowly for θ (h) >0.5. This type of entropy measures such as E_g3 (h) is called conservative. The graph of E_g4 (h) is similar to E_g3 (h), so E_g4 (h) also belongs to this type. Compared with E_g3 (h), E_g2 (h) has the opposite characteristic. When θ (h) closes to 0.5, the value of E_g2 (h) increases quickly for θ (h) ≤0.5 and decreases quickly for θ (h) >0.5. Simultaneously, when θ (h) stays away from 0.5, the value of E_g2 (h) increases slowly for θ (h) ≤0.5 and decreases slowly for θ (h) >0.5. This type of entropy measures such as E_g2 (h) is called risker. Regarding E_g1 (h), its graph is a solid line which increases and decreases steadily. This type of entropy measure, E_g1 (h), is called independent. Therefore, different type of entropy measures can be chosen according to the attitude of a decision-maker for the value θ (h).

3.3 Entropy measures for HFSs

HFEs are the basic elements of HFSs. According to the entropy measures of HFEs, the entropy axiomatic definition and some concrete entropy formulas for HFSs are proposed.

Definition 4. A real-valued function E: HFS(X)→[0, 1] is called an entropy measure for HFSs, if it satisfies the following axiomatic requirements.

Let A = {〈x_i, h_A (x_i) 〉|x_i ∈ X} and B = {〈x_j, h_B (x_j) 〉|x_j ∈ X} be two HFSs,

E (A) =0 if and only if h_A (x_i) = {0} or h_A (x_i) = {1} for all x_i ∈ X;

E (A) =1 if and only if θ (h_A (x_i)) =0.5 for all x_i ∈ X;

E (A) ≤ E (B), if θ (h_A (x_i)) ≤ θ (h_B (x_i)) and η (h_A (x_i)) ≤ η (h_B (x_i)) for θ (h_B (x_i)) ≤0.5, or θ (h_A (x_i)) ≥ θ (h_B (x_i)) and η (h_A (x_i)) ≤ η (h_B (x_i)) for θ (h_B (x_i)) ≥0.5 for any x_i ∈ X;

E (A) = E (A^c), where A^c is the complement of an HFS A on X, defined by $A^{c} = {〈 x_{i}, h_{A}^{c} (x_{i}) 〉 | x_{i} \in X}$ .

Theorem 3. Let A = {〈x_i, h_A (x_i) 〉|x_i ∈ X} be an HFS on X. Then $\overline{E} (A) = \frac{1}{n} \sum_{i = 1}^{n} E (h_{A} (x_{i})),$ (12) is an entropy measure for the HFS A, where E (·) is an entropy measure for HFEs.

Proof. Since E satisfies the requirements in Definition 3, it is easy to show that $\overline{E}$ satisfies the requirements in Definition 4.

From Theorems 2 and 3, for HFS A = {〈x_i, h_A (x_i) 〉|x_i ∈ X}, a family of entropy measures is obtained: ${\overline{E}}_{g} (A) = \frac{1}{n} \sum_{i = 1}^{n} \frac{f (θ (h (x_{i}))) + k η (h (x_{i}))}{1 + k η (h (x_{i}))},$ (13) where k ∈ [0, 1] and the function f (x) : [0, 1] → [0, 1] satisfies f (1 - x) = f (x), and is strictly monotone increasing with respect to x ∈ (0, 0.5] and strictly monotone decreasing with respect to x ∈ [0.5, 1). Besides, it interpolates three points, (0, 0), $(\frac{1}{2}, 1)$ and (1, 0). Taking some special functions f (x) such as the ones used in Subsection 3.2. for HFEs, we can get different entropy measuresfor HFSs.

4 Comparisons with existing entropy axiomatic definitions and entropy measures for HFEs

This section introduces a comparison between the proposed axiomatic Definition 3 and the entropy measures with those presented in [9] and [32]. Several examples are shown to illustrate that the proposed entropy measures can reflect both fuzziness and hesitation of HFEs.

4.1 Theoretical analysis

Xu and Xia [32] proposed the following axiomatic definition and entropy formulas for HFEs.

Definition 5. [32] An entropy on an HFE h is a real-valued function E : H → [0, 1], satisfying the following axiomatic requirements:

E (h) =0, if and only if h = {0} or h = {1};

E (h) =1, if and only if hⁱ+ h^l-i+1= 1 for i = 1, 2, …, l;

E (h₁) ≤ E (h₂), if $h_{1}^{i} \leq h_{2}^{i}$ for $h_{2}^{i} + h_{2}^{l - i + 1} \leq 1$ , or $h_{1}^{i} \geq h_{2}^{i}$ for $h_{2}^{i} + h_{2}^{l - i + 1} \geq 1$ , where h₁ and h₂ have the same length l obtained by repeating elements.

E (h) = E (h^c).

Based on Definition 5, the following entropy measures are proposed: $\begin{matrix} E_{1} (h) = & \frac{1}{l (\sqrt{2} - 1)} \sum_{i = 1}^{l} {sin \frac{π (h^{i} + h^{l - i + 1})}{4} \\ + sin \frac{π (2 - h^{i} - h^{l - i + 1})}{4} - 1}, \end{matrix}$ (14) $\begin{matrix} E_{2} (h) = & \frac{1}{l (\sqrt{2} - 1)} \sum_{i = 1}^{l} {cos \frac{π (h^{i} + h^{l - i + 1})}{4} \\ + cos \frac{π (2 - h^{i} - h^{l - i + 1})}{4} - 1}, \end{matrix}$ (15) $\begin{matrix} E_{3} (h) = & - \frac{1}{l ln 2} \sum_{i = 1}^{l} {\frac{h^{i} + h^{l - i + 1}}{2} ln \frac{h^{i} + h^{l - i + 1}}{2} \\ + \frac{2 - h^{i} - h^{l - i + 1}}{2} \ln \frac{2 - h^{i} - h^{l - i + 1})}{2}} \end{matrix}$ (16) $\begin{matrix} E_{4} (h) = & \frac{1}{l (2^{(1 - s) t} - 1)} \sum_{i = 1}^{l} {((\frac{h^{i} + h^{l - i + 1}}{2})^{s} \\ + (\frac{2 - h^{i} - h^{l - i + 1}}{2})^{s})^{t} - 1} . \end{matrix}$ (17)

Definition 5 generalizes the axiomatic definition of entropy for FSs [6] and aims to define the entropy measure of an HFE h based on the similarity degree of h and h^c. It is noted that the condition (E3) is too strong to discriminate the uncertain degrees of some HFEs, even though they are apparently different. While using (E3) of the Definition 3, the entropy values of more HFEs can be compared and this comparison is more coherent with people’s intuition.

For example, suppose that the membership degree of one element to a set provided by one decision group is represented by the HFE h₁ = {0.1, 0.9} and the membership degree provided by another decision group is represented by the HFE h₂ = {0.11}.

The HFE h₁ = {0.1, 0.9} depicts the situation that some experts in the group provide the membership degree 0.1, and the other provide the membership degree 0.9. There are bigger disagreements among experts in this group. The information provided by the HFE h₁ = {0.1, 0.9} involves hesitation and we are not sure whether the element belongs to the set or not. The HFE h₂ = {0.11} implies that the element is belonging to the set with a degree 0.11. The information does not involve hesitation. Intuitively, the uncertain degree of h₁ is higher than h₂. But for the two HFEs h₁ = {0.1, 0.9} and h₂ = {0.11, 0.11}, we have 0.11 > 0.1 and 0.11 < 0.9. So, from condition (E3) of the Definition 5, it is not possible to compare the entropies of the HFEs h₁ and h₂. However, taking into account (E3) of the proposed Definition 3, we have E (h₁) > E (h₂) since θ (h₂) < θ (h₁) ≤0.5 and η (h₂) < η (h₁). This result accords with people’s intuition.

On the other hand, using the condition (E3) of the Definition 5 to compare two HFEs, it is necessary to extend the shorter HFE to have the same length as the longer one. Any value in the shorter one can be added to extend it. The comparison results will be susceptible to the added elements. However, if (E3) of the proposed Definition 3 is used, it is not necessary to add any value to compare the HFEs.

For example, let h₁ = {0.2, 0.5} and h₂ = {0.2, 0.3, 0.4} be two HFEs. If h₁ is extended to h₁ = {0.2, 0.5, 0.5}, then E (h₁) > E (h₂) using the condition (E3) of the Definition 5. But if it is extended to h₁ = {0.2, 0.2, 0.5}, then it is not possible to compare the two HFEs by this condition. Nevertheless, by using the condition (E3) of the Definition 5, since θ (h₂) < θ (h₁) ≤0.5 and η (h₂) < η (h₁), then E (h₁) > E (h₂).

The entropy measures E₁, E₂, E₃ and E₄ defined by Equations 14–17 which consider the fuzziness of the HFEs are analyzed. In fact, for an HFE h = {h¹, h², …, h^l}, the entropy measures change with the values of $\frac{h^{i} + h^{l - i + 1}}{2}$ . If $\frac{h^{i} + h^{l - i + 1}}{2} < 0.5$ , the bigger the value of $\frac{h^{i} + h^{l - i + 1}}{2}$ , then the bigger the value of θ (h) is, and furthermore the bigger the entropy is. If $\frac{h^{i} + h^{l - i + 1}}{2} > 0.5$ , the bigger the value of $\frac{h^{i} + h^{l - i + 1}}{2}$ , then the bigger the value of θ (h) is, and the smaller the entropy is. For any two HFEs h₁ and h₂ with the same length l, if $h_{1}^{i} + h_{1}^{l - i + 1} = h_{2}^{i} + h_{2}^{l - i + 1}$ for any i = 1, 2, …, l, then θ (h₁) = θ (h₂), thus E (h₁) = E (h₂). In contrast to Equations 14–17, the entropy calculated by Equation 7 changes with the value of score function and the value of deviation function of an HFE. It considers not only the fuzziness of an HFE, but also its hesitation degree. Therefore, the entropy measure defined by Equation 7 has bigger ability to compare the uncertainties of HFEs.

On the other hand, Farhadinia [9] gave the following axiomatic definition of entropy measure based on the distance d proposed in [31] between two HFEs.

Definition 6. An entropy on an HFE h is a real-valued function E : H → [0, 1], satisfying the following axiomatic requirements:

E (h) =0, if and only if h = {0} or h = {1};

E (h) =1, if and only if h = {0.5};

E (h₁) ≤ E (h₂), if d (h, {0.5}) ≥ d (h, {0.5})

E (h) = E (h^c).

Based on Definition 6, the following theorem is given in [9] to provide an approach to build a family of entropies for HFEs using a distance between HFEs.

Theorem 4. Let Z : [0, 1] → [0, 1] be a strictly monotone decreasing real function, and d be a distance between HFEs. Then $E_{d} (h) = \frac{2 d (h, {0.5}) - Z (1)}{Z (0) - Z (1)}$ is an entropy for the HFE h based on the corresponding distance d.

Xu and Xia [31] defined a variety of distance measures for HFSs that have been reviewed in Section 2. These distance measures can be used to calculate the distance between an HFE h and {0.5}. Therefore, in Theorem 4, if Z (x) =1 - x and d is defined by Equation 2 with λ = 1. Then $E_{d} (h) = 1 - 2 d_{gh} (h, {0.5}) = 1 - \frac{1}{l} \sum_{i = 1}^{l} 2 | h^{i} - 0.5 |$ (18)

Note that Definition 6 is based on the distance d between the HFEs h and {0.5}, and it only considers the fuzziness of an HFE. Therefore, for any two HFEs that have the same distance to {0.5}, the entropy is equal by using the entropy formula E_d defined by Equation 18. For example, let h₁ = {0, 1}, h₂ = {0} be two HFEs. The entropy obtained for them is E_d (h₁) = E_d (h₂) =0. It is easy to see that the result is not consistent with our reasoning. In fact, h₁ = {0, 1} represents that one expert is absolutely in favor and the other one is absolute opposition, and h₂ = {0} represents that the two experts are absolute opposition. The uncertainty of information represented by h₁ = {0, 1} should be the biggest. But using the Equation 18, we obtain E_d (h₁) = E_d (h₂) =0, which is a shortcoming of the entropy measure defined in [9]. Moreover, from the above example, it can be seen that the entropy measure E_d defined in Theorem 4 does not meet the condition (E1) of the Definition 6.

4.2 Comparisons with examples

This subsection carries out a further comparison between the existing entropy formulas by means ofan example.

Example 2. Let h₁ = {0.2, 0.4}, h₂ = {0.3, 0.5} , h₃ = {0.3, 0.4, 0.5}, h₄ = {0.4, 0.5}, h₅ = {0.3, 0.6}, h₆ = {0.3, 0.5, 0.6}, h₇ = {0.2, 0.5, 0.7} , h₈ = {0.4, 0.5, 0.6}, h₉ = {0, 1}, be nine HFEs.

The entropies of these HFEs are calculated by the entropy measures E₁, E₂, E₃, E₄, E_g1, E_g2,E_g3,E_g4 and E_d (λ = 1), respectively. The results are shown in Table 1.

From Table 1, we can see that entropy values calculated by E_i (i = 1, 2, 3, 4) and E_gi (i = 1, 2, 3, 4) get larger when the score function values θ (h) increase. But the entropies E₁, E₂, E₃ and E₄ cannot distinguish the uncertainty of HFEs that have the same score function values, since E_i (h₆) = E_i (h₇) (i = 1, 2, 3, 4) and E_i (h₈) = E_i (h₉) (i = 1, 2, 3, 4). The HFEs h₆ and h₇, h₈ and h₉ have the same score function values and different deviation values, respectively. The dispersion degree of elements in h₇ is greater than in h₆, and the dispersion degree of elements in h₉ is greater than in h₈. Obviously, h₇ is intuitively more uncertain than h₆, and h₉ is more uncertain than h₈. Using the proposed entropies E_gi (i = 1, 2, 3, 4), it is obtained that E_gi (h₆) < E_gi (h₇) (i = 1, 2, 3, 4), that it is consistent with human beings reasoning.

There is also some inconsistent results using E_d defined by Equation 18. The entropy E_d (h₂) = E_d (h₃) and E_d (h₄) > E_d (h₅), while E_gi (h₃) < E_gi (h₂) and E_gi (h₄) < E_gi (h₅) (i = 1, 2, 3, 4). Therefore, the results E_d (h₂) = E_d (h₃) and E_d (h₄) > E_d (h₅) are not consistent with our reasoning. In fact, h₂ and h₃, h₄ and h₅ have the same score function values respectively. The dispersion degree of elements in h₂ is greater than in h₃, and the dispersion degree of elements in h₅ is greater than in h₄. Obviously, h₂ is intuitively more uncertain than h₃ and h₅ is more uncertain than h₄, that it is the fact reflected by the entropies E_gi (i = 1, 2, 3, 4).

From the above analysis, the entropy measures E_{g
_i} (i = 1, 2, 3, 4) are more effective to reflect hesitation and fuzziness of HFEs.

5 A method for multi-criteria decision-making based on entropy measures for HFEs

Entropy measures have been applied in many problems such as optimizing the distinguishability of input space partitioning and assessing the weights of experts or criteria in intuitionistic fuzzy decision-making [11 , 34]. In this section, a method is proposed to solve multi-criteria decision-making problems with unknown criterion weights [5]. The entropy measures for HFSs are used to determinate the criteria’ weights and the idea of TOPSIS method is used to rank alternatives.

The multi-criteria decision-making problem which is considered in this paper can be represented as follows. There are m alternatives, denoted by X = {x₁, x₂, …, x_m}. Each alternative is assessed by means of n criteria, denoted by C = {C₁, C₂, …, C_n}. Assume that the weights of the criteria C_j(j = 1, 2, …, n) are unknown. The characteristics of the alternative x_i in terms of the criterion C_j are represented by the followingHFSs: $M_{i} = {〈 C_{j}, h_{ij} 〉 | C_{j} \in C}, i = {1, 2, \dots, m},$ where h_ij is an HFE that indicates the degree in which the alternative M_i satisfies the criterion C_j. Espert’s goal is to obtain a ranking of alternatives.

In practical multi-criteria decision-making problems, it is an important research topic to determine the weights of criteria. In the following, it is proposed a method to obtain the weights of criteria based on the proposed entropy measures according to experts’ evaluation information, and a method to solve the above multi-criteria decision-makingproblem.

The multi-criteria decision-making approachcontains three processes: (1) determinate the weights of the criteria; (2) derive the comprehensive evaluations of the alternatives; (3) rank the alternatives.

Determination of the weights of the criteria: the proposed entropy measures are used to compute the criteria weights. Let C_j = {〈x_i, h_ij〉|x_i ∈ X} , j = {1, 2, …, n} be an HFS on the alternative set X, which includes the overall assessment values for all the alternatives x_i (i = 1, 2, …, m) under the criteria C_j. Therefore, the entropy $\overline{E}$ _j of C_j can be calculated by Equations 8–11, where ${\overline{E}}_{j} = \frac{1}{m} \sum_{i = 1}^{m} E (h_{ij})$ . ${\overline{E}}_{j}$ indicates the uncertainty degree of the assessment provided for the criterion C_j. During the practical decision-making process, we usually expect that the uncertainty degree of the assessment is as small as possible. Thus, if the entropy value ${\overline{E}}_{j}$ related to the criterion C_j is lower, we assign it a higher weight, and vice versa. Therefore, the criteria weights are defined as follows: $ω_{j} = \frac{1 - {\overline{E}}_{j}}{n - \sum_{j = 1}^{n} {\overline{E}}_{j}}, j = {1, 2, \dots, n}$ (19)

Deriving the comprehensive evaluations of the alternatives: this process is divided into three steps.

Firstly it is necessary to find the positive-ideal solution and negative-ideal solution. Let J₁ and J₂ be the sets of benefit criteria and cost criteria in the criterion set C, respectively. Suppose that $H^{+} = {〈 C_{j}, h_{j}^{+} 〉 | C_{j} \in C}$ is the hesitant fuzzy positive-ideal solution, and $H^{-} = {〈 C_{j}, h_{j}^{-} 〉 | C_{j} \in C}$ is the hesitant fuzzy negative-ideal solution, where $h_{j}^{+} = {1}$ , $h_{j}^{-} = {0}$ , j ∈ J₁ and $h_{j}^{+} = {0}$ , $h_{j}^{-} = {1}$ , j ∈ J₂.

By using Equation 3, the distance between the alternative M_i and the positive-ideal solution or the negative-ideal solution can be computed:

$\begin{matrix} D^{+} (M_{i}) = d_{ghw} (M_{i}, H^{+}), \\ D^{-} (M_{i}) = d_{ghw} (M_{i}, H^{-}), \end{matrix}$

being i = {1, 2, … m} .

The relative closeness degree D (M_i) of the alternative M_i to the ideal solution is obtained as follows. $D (M_{i}) = \frac{D^{+} (M_{i})}{D^{+} (M_{i}) + D^{-} (M_{i})}, i = 1, 2, \dots m .$

Ranking the alternatives: the alternatives are ordered according to the relative closeness degrees.

The following steps show how to apply the multi-criteria decision making method.

Calculate the criterion weights by Equation 19.

Calculate the distances of each alternative to the positive-ideal solution D⁺(M_i), and the negative-ideal solution D^-(M_i).

Calculate the relative closeness degree D (M_i) for each alternative.

Rank the alternatives M_i according to the values of D (M_i) (i = 1, 2, …, m) in ascending order, and the smaller the value of D (M_i), the better the alternative M_i.

6 Illustrative example

A case study concerning the health-care waste management is employed to illustrate the efficiency of the multi-criteria decision making method proposed in the previous section.

Nowadays, the competition among telecommunications services is increasing and it is much more difficult for SMEs (Small and Medium-sized Enterprises) to choose a suitable telecommunications service to improve their business operations, since ample resources can be a big obstacle. Suppose that a SME has to select the best telecommunications service provider to improve its benefits. There are five possible alternatives: provider 1 (M₁), provider 2 (M₂), provider 3 (M₃), provider 4 (M₄) and provider 5 (M₅). Based on the society research, four major criteria are considered to evaluate these five telecommunications service providers. These criteria are: The satisfaction of price (C₁), Quality (C₂), Service (C₃), and Safeguard (C₄). A detailed description of such criteria is given inTable 2.

Let us suppose a decision organization with five experts authorized to assess the satisfactory degree of an alternative with respect to a criterion, which is represented by an HFE. The evaluations of the five possible alternatives M_i (i = 1, 2, …, 5) under the above four criteria can be represented by the following HFSs:

{〈C₁, {0.5, 0.6} 〉, 〈C₂, {0.6, 0.7} 〉, 〈C₃, {0.3, 0.4} 〉, 〈C₄, {0.2, 0.3} 〉},

{〈C₁, {0.6, 0.7, 0.8} 〉, 〈C₂, {0.7, 0.8} 〉, 〈C₃, {0.7, 0.8} 〉, 〈C₄, {0.4, 0.5} 〉},

{〈C₁, {0.5, 0.6, 0.7} 〉, 〈C₂, {0.5, 0.6} 〉, 〈C₃, {0.5, 0.6, 0.7} 〉, 〈C₄, {0.6, 0.7} 〉},

{〈C₁, {0.8, 0.9} 〉, 〈C₂, {0.6, 0.7} 〉, 〈C₃, {0.3, 0.4, 0.5, 0.6} 〉, 〈C₄, {0.2, 0.3, 0.4} 〉},

{〈C₁, {0.6, 0.7, 0.8} 〉, 〈C₂, {0.4, 0.5, 0.6, 0.7} 〉, 〈C₃, {0.7, 0.8, 0.9} 〉, 〈C₄, {0.5, 0.6, 0.7} 〉} .

The multi-criteria decision making approach proposed in Section 5 will be used to get the most desirable alternative(s).

Determine the criterion weights.

With the entropy measures E_g1, E_g2, E_g3 and E_g4 defined by Equations 8–11, are calculated the entropies of HFEs, which construct the entropy matrices shown in Table 3, respectively.

According to each entropy matrix calculated by E_gi and Equation 19, the entropies ${\overline{E}}_{i}$ and the weights w_i of criteria C_i (i = 1, 2, 3, 4) are calculated and shown inTable 4.

Calculate the distance D⁺(M_i) and D^-(M_i) between each alternative and the positive-ideal and negative-ideal solution, respectively. Afterwards, the relative closeness degrees D (M_i), for each alternative is obtained (see Table 5).

Taking into account the relative closeness degree obtained for each alternative D (M_i) (i = 1, 2, …, m), the ranking of alternatives obtained by using different entropy measures E_gi (i = 1, 2, 3, 4) is as follows: $M_{1} ≻ M_{4} ≻ M_{3} ≻ M_{2} ≻ M_{5} .$

In the above calculation process, we adopt the entropy formulas E_g1, E_g2, E_g3 and E_g4 to calculate the criteria weights and the relative closeness degrees of alternatives, respectively. From Table 4, it is easy to see that criteria weights obtained by each entropy measure are different, but the ranking of these weights are the same, that is, w₁ > w₃ > w₄ > w₂. Under different entropy measures, the relative closeness degrees of alternatives are different, but the gaps between these values are very small, therefore the ranking of alternatives is the same. The comparisons of weights and relative closeness degrees under the four entropy measures are shown inFigs. 2 and 3.

7 Conclusions

An HFS, whose membership is represented by a set of possible values, is more suitable to represent the uncertain information when experts hesitate among several values. In this paper, the entropy measures (being important topics of information measures) of HFEs and HFSs have been studied. An axiomatic definition and a concrete entropy formula for HFEs have been proposed. Then a family of entropy measures based on the entropy measure aforementioned has been presented. These entropy measures have been compared with the existing ones. The comparison results reflect that the proposed axiomatic definitions and entropy measures can depict both fuzziness and hesitation of HFEs and HFSs, and they are able to compare HFEs when the existing ones cannot. Finally, the proposed entropy measures have been used to determinate the criteria weights and a multi-criteria decision making approach has been developed to deal with hesitant fuzzy information. In the future, we will further investigate the entropy theory on the hesitant fuzzy linguistic term sets and its application in decision making.

Footnotes

Acknowledgments

The work was partly supported by the National Natural Science Foundation of China (71371107), the National Science Foundation of Shandong Province (ZR2013GM011), the Spanish National research project TIN2015-66524-P, Spanish Ministry of Economy and Finance Postdoctoral Training (FPDI-2013-18193) and ERDF.

References

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets and Syst20 (1986), 87–96.

Atanassov

, More on intuitionistic fuzzy sets, Fuzzy Sets and Syst33 (1989), 37–46.

Bustince

, Montero

, Pagola

, Barrenechea

and Gomez

, Handbook of Granular Computing, A survey of interval-valued fuzzy sets, John Wiley & Sons, Ltd, 2008, pp. 489–515.

Cevik Onar

, Oztaysi

and Kahraman

, Strategic decision selection using hesitant fuzzy TOPSIS and interval type-2 fuzzy AHP: A case study, International Journal of Computational Intelligence Systems7(5) (2014), 1002–1021.

Kahraman

and Cevik Onar

and Oztaysi

, Fuzzy multicriteria decision-making: A literature review, International Journal of Computational Intelligence Systems8(4) (2015), 637–666.

De Luca

and Termini

, A definition of nonprobabilistic entropy in the setting of fuzzy sets theory, Information and Control20 (1972), 301–312.

Dubois

and Prade

, Fuzzy sets and systems: Theory and applications, New York, Academic Press, 1980.

Fan

J.L.

, Some new fuzzy entropy formulas, Fuzzy Sets and Syst128 (2002), 277–284.

Farhadinia

, Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets, Inform Sciences240 (2013), 129–144.

10.

Gau

W.L.

and Buehrer

D.J.

, Vague sets,, IEEE Transactions on Systems, Man and Cybernetics23(2) (1993), 610–614.

11.

Liang

and Wei

C.P.

, An Atanassov’s intuitionistic fuzzy multi-attribute group decision making method based on entropy and similarity measure,, International Journal of Machine Learning and Cybernetics5 (2014), 435–444.

12.

Liao

H.C.

and Xu

Z.S.

, Extended hesitant fuzzy hybrid weighted aggregation operators and their application in decision making, Soft Computing19(9) (2015), 2551–2564.

13.

Liao

H.C.

and Xu

Z.S.

, Subtraction and division operations over hesitant fuzzy sets, Journal of Intelligent & Fuzzy Systems27(1) (2014), 65–72.

14.

Liao

H.C.

, Xu

Z.S.

and Zeng

X.J.

, Novel correlation coefficients between hesitant fuzzy sets and their application in decision making,, Knowl-Based Syst82 (2015), 115–127.

15.

Liao

H.C.

and Xu

Z.S.

, Approaches to manage hesitant fuzzy linguistic information based on the cosine distance and similarity measures for HFLTSs and their application in qualitative decision making, Expert Systems with Applications42(12) (2015), 5328–5336.

16.

Liao

H.C.

, Xu

Z.S.

, Zeng

X.J.

and Merigó

J.M.

, Qualitative decision making with correlation coefficients of hesitant fuzzy linguistic term sets, Knowl-Based Syst76 (2015), 127–138.

17.

Mendel

J.M.

, Uncertain rule-based fuzzy logic systems: Introduction and new directions, Prentice-Hall PTR, 2001.

18.

Miyamoto

, Remarks on basics of fuzzy setsand fuzzy multisets, Fuzzy Sets and Syst156 (2005), 427–431.

19.

Parkash

, Sharma

P.K.

and Mahajan

, New measures of weighted fuzzy entropy and their applications for the study of maximum weighted fuzzy entropy principle, Inform Sciences178 (2008), 2389–2395.

20.

Rodríguez

R.M.

, Martínez

and Herrera

, Hesitant fuzzy linguistic term sets for decision making, IEEE Transactions on Fuzzy Systems20(1) (2012), 109–119.

21.

Rodríguez

R.M.

, Martínez

, Torra

, Xu

Z.S.

and Herrera

, Hesitant fuzzy sets: State of the art and future directions, International Journal of Intelligent Systems29(6) (2014), 495–524.

22.

Shang

X.G.

and Jiang

W.S.

, A note on fuzzy information measures,, Pattern Recognition Letter18 (1997), 425–432.

23.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems25 (2010), 529–539.

24.

Torra

and Narukawa

, On hesitant fuzzy sets and decision, The 18th IEEE Int Conf Fuzzy Systems, Jeju Island, Korea, 2009, pp. 1378–1382.

25.

Wei

C.P.

and Tang

X.J.

, An intuitionistic fuzzy group decisionmaking approach based on entropy and similarity measures, International Journal of Information Technology & Decision Making10(6) (2011), 1111–1130.

26.

Wei

C.P.

, Ren

and Rodríguez

R.M.

, A hesitant fuzzy linguistic TODIM method based on a score function, International Journal of Computational Intelligence Systems8(4) (2015), 701–712.

27.

Wei

G.W.

and Zhang

, A multiple criteria hesitant fuzzy decision making with Shapley value-based VIKOR method, Journal of Intelligent and Fuzzy Systems26(2) (2014), 1065–1075.

28.

Wei

G.W.

, Wang

H.J.

, Zhao

X.F.

and Lin

, Approaches to hesitant fuzzy multiple attribute decision making with incomplete weight information, Journal of Intelligent and Fuzzy Systems26(1) (2014), 259–266.

29.

Wei

G.W.

, Hesitant fuzzy prioritized operators and their application to multiple attribute group decision making, Knowl-Based Syst31 (2012), 176–182.

30.

Xia

M.M.

and Xu

Z.S.

, Hesitant fuzzy information aggregation in decision making, International Journal of Approximate Reasoning52(3) (2011), 395–407.

31.

Z.S.

and Xia

M.M.

, Distance and similarity measures for hesitant fuzzy sets, Inform Sciences181(11) (2011), 2128–2138.

32.

Z.S.

and Xia

M.M.

, Hesitant fuzzy entropy and crossentropy and their use in multiattribute decision-making, International Journal of Intelligent Systems27 (2012), 799–822.

33.

Yager

R.R.

, On the measure of fuzziness and negation. Part 1: Membership in the unit interval, International Journal of General Systems5 (1979), 221–229.

34.

, Fuzzy decision-making method based on the weighted correlation coefficient under intuitionistic fuzzy environment, European Journal of Operational Research205 (2010), 202–204.

35.

Zadeh

L.A.

, Fuzzy sets, Information and Control8(3) (1965), 338–356.