Note on the Choquet Integral as an Interval-Valued Aggregation Operators and Their Applications

Abstract

The concept of an interval-valued capacity is motivated by the goal to generalize a capacity, and it can be used for representing an uncertain capacity. In this paper, we define the discrete interval-valued capacities, a measure of the entropy of a discrete interval-valued capacity, and, Choquet integral with respect to a discrete interval-valued capacity. In particular, we discuss the Choquet integral as an interval-valued aggregation operator and discuss an application of them.

1. Introduction

Let (𝑋,Ω) be a measurable space. A capacity (or a fuzzy measure) on 𝑋 is a nonnegative monotone set function 𝜇∶Ω→ℝ⁺=[0,∞] with 𝜇(∅)=0. Many researchers have been studying a discrete capacity in many topics such as capacity functionals of random sets (see [1–5]) and entropy-like measures (see [6–9]).

The Choquet integral with respect to a capacity of a nonnegative measurable function 𝑓 is given by𝒞_𝜇𝑓=(𝐶)𝑓𝑑𝜇=^∞₀𝜇_𝑓(𝛼)𝑑𝛼, (1.1) where 𝜇_𝑓(𝛼)=𝜇({𝑥∈𝑋∣𝑓(𝑥)>𝛼}) and the integral on the right-hand side is an ordinary one. If we take 𝑁={1,2,…,𝑛} and 𝑓∶𝑁→ℝ⁺ by 𝑓(𝑖)=𝑥_𝑖 for all 𝑖∈𝑁, then we have𝒞_𝜇𝑓=^𝑛_𝑖=1𝑥_𝜋(𝑖)𝜇𝐴_𝜋(𝑖)−𝜇𝐴_𝜋(𝑖+1), (1.2) where 𝜋 is a permutation on {1,2,…,𝑛} such that 𝑥_𝜋(1)≤𝑥_𝜋(2)≤⋯≤𝑥_𝜋(𝑛) and 𝐴_𝜋(𝑖)={𝜋(𝑖),…,𝜋(𝑛)} and 𝐴_𝜋(𝑛+1)=∅(see [1–4, 6, 9, 10]). Note that if we put 𝑥_𝜋(0)=0 and 𝐴_𝜋(𝑛+1)=∅, then we obtain the following formula:𝒞_𝜇𝑓=^𝑛−1_𝑖=0𝑥_𝜋(𝑖+1)−𝑥_𝜋(𝑖)𝜇𝐴_𝜋(𝑖+1). (1.3)

By using interval-valued functions to express uncertain functions, we have studied the Choquet integral with respect to a capacity of an interval-valued function which is able to better handle the representation of decision making and information theory (see [10–16]). During the last decade, it has been suggested to use intervals in order to represent uncertainty in the area of decision theory and information theory, for example, calculation of economic uncertainty [8], theory of interval probability as a unifying concept for uncertainty [17], and the Choquet integral of uncertain functions [3, 12–16, 18]. Recently, Xu et al. [19–24] have been studying the application of the Choquet integral with uncertain and fuzzy information.

The main idea of this paper is to use the concept of an interval-valued capacity in the entropy-like measure which is an aggregation defined by the discrete interval-valued capacities. In Section 2, we introduce the Choquet integral with respect to an interval-valued capacity and discuss some of its properties. In Section 3, we investigate the interval-valued weighted arithmetic mean, the interval-valued Shannon entropy, the interval-valued weighted averaging operator, and an interval-valued measure of the entropy of an interval-valued capacity. In Section 4, we give the problem of evaluating students as an example where interval-valued weights and some suitable interval-valued capacity are used in practical situation. In Section 5, we give a brief summary results and some conclusions.

2. The Choquet Integral with Respect to a Discrete Interval-Valued Capacity

Throughout this paper, 𝐼(ℝ⁺) is the set of all closed intervals in ℝ⁺=[0,∞), that is, 𝐼ℝ⁺=𝑎₁,𝑎₂∣𝑎₁,𝑎₂∈ℝ⁺,𝑎₁≤𝑎₂. (2.1) For any 𝑎∈ℝ⁺, we define 𝑎=[𝑎,𝑎]. Obviously, 𝑎∈𝐼(ℝ⁺)(see [13–15]).

Definition 2.1. If 𝑎=[𝑎₁,𝑎₂],𝑏=[𝑏₁,𝑏₂]∈𝐼(ℝ⁺), and 𝑘∈ℝ⁺, then one defines arithmetic, minimum, order, and inclusion operations as follows:(1)𝑎+𝑏=[𝑎₁+𝑏₁,𝑎₂+𝑏₂],(2)𝑘𝑎=[𝑘𝑎₁,𝑘𝑎₂],(3)𝑎𝑏=[𝑎₁𝑏₁,𝑎₂𝑏₂],(4)𝑎∧𝑏=[𝑎₁∧𝑏₁,𝑎₂∧𝑏₂],(5)𝑎∨𝑏=[𝑎₁∨𝑏₁,𝑎₂∨𝑏₂],(6)𝑎≤𝑏 if and only if 𝑎₁≤𝑏₁ and 𝑎₂≤𝑏₂,(7)𝑎<𝑏 if and only if 𝑎≤𝑏 and 𝑎≠𝑏, (8)𝑎⊂𝑏 if and only if 𝑏₁≤𝑎₁ and 𝑎₂≤𝑏₂.

Let 𝑈 be a countably infinite set as the universe of discourse and 𝒫(𝑈) the power set of 𝑈. We propose an interval-valued capacity and discuss some of its properties.

Definition 2.2. (1) An interval-valued set function 𝜇=[𝜇_𝑙,𝜇_𝑟]∶𝒫(𝑈)→𝐼(ℝ⁺) is said to be a discrete interval-valued capacity on 𝑈 if it satisfies the following conditions:(i)𝜇(∅)=0, (ii)𝜇(𝑆)≤𝜇(𝑇), whenever 𝑆,𝑇∈𝒫(𝑈) and 𝑆⊂𝑇.
(2) A set 𝐷(∈𝒫(𝑈)) is said to be a carrier (or support) of an interval-valued capacity 𝜇 if 𝜇(𝑆)=𝜇(𝐷∩𝑆) for all 𝑆∈𝒫(𝑈).
(3) An interval-valued capacity 𝜇 with nonempty finite carrier 𝐷(∈𝒫(𝑈)) is said to be normalized if 𝜇(𝐷)=1.

For any integer 𝑘≥1, the set {1,…,𝑘} will simply be denoted by [𝑘] and 𝐼([0,1])={[𝑎₁,𝑎₂]∣𝑎₁,𝑎₂∈[0,1]and𝑎₁≤𝑎₂}. For the sake of convenience, we will henceforth assume that 𝐷=𝑁 is the 𝑛-element set [𝑛]. We denote by IVC the set of interval-valued capacities with a nonempty finite carrier on 𝑈 and by IVC_𝑁 the set of normalized interval-valued capacities having 𝑁⊂𝑈 as a nonempty finite carrier.

Definition 2.3. (1) An interval-valued capacity 𝜇∈IVC_𝑁 is said to be additive if 𝜇(𝑆∪𝑇)=𝜇(𝑆)+𝜇(𝑇) for all disjoint subsets 𝑆,𝑇⊂𝑁.
(2) 𝜇∈IVC_𝑁 is said to be cardinality based if for all 𝑇⊂𝑁, 𝜇(𝑇) depends only on the cardinality of 𝑇; that is, there exists 𝜇₀,𝜇₁,…,𝜇_𝑛∈𝐼([0,1]) such that 𝜇(𝑇)=𝜇_𝑡=[𝜇_𝑙𝑡,𝜇_𝑟𝑡] for all 𝑇⊂𝑁 such that |𝑇|=𝑡, where |𝑇| is the cardinality of 𝑇.

In [6, p. 135], there is only one normalized capacity 𝜇^∗_𝑁 with a nonempty finite carrier 𝑁 which is both additive and cardinality based, and in this case, 𝜇^∗_𝑁 is given by 𝜇^∗_𝑁(𝑇)=𝑡/𝑛 for all 𝑇⊂𝑁 such that |𝑇|=𝑡. Thus we can obtain the following theorem.

Theorem 2.4. If 𝜇∈IVC_𝑁 is both additive and cardinality based, then 𝜇(𝑇)=[𝑡/𝑛,𝑡/𝑛], for all 𝑇⊂𝑁 with |𝑇|=𝑡.

Theorem 2.4 implies that if a discrete interval-valued normalized capacity 𝜇 is both additive and cardinality based, then it is a discrete real-valued capacity (or a real-valued monotone set function). By Definition 2.3, we can easily obtain the following theorem.

Theorem 2.5. (1) An interval-valued set function 𝜇=[𝜇_𝑙,𝜇_𝑟] is a discrete interval-valued capacity if and only if 𝜇_𝑙 and 𝜇_𝑟 are discrete capacities.(2) A set 𝑁(⊂𝑈) is a carrier of 𝜇=[𝜇_𝑙,𝜇_𝑟] if and only if 𝑁 is a carrier of both 𝜇_𝑙 and 𝜇_𝑟.(3)𝜇=[𝜇_𝑙,𝜇_𝑟] is normalized if and only if 𝜇_𝑙 and 𝜇_𝑟 are normalized.(4)𝜇=[𝜇_𝑙,𝜇_𝑟] is additive if and only if 𝜇_𝑙 and 𝜇_𝑟 are additive.(5)𝜇=[𝜇_𝑙,𝜇_𝑟] is cardinality based if and only if 𝜇_𝑙 and 𝜇_𝑟 are cardinality based.

By using formula (1.3) of the Choquet integral and a discrete interval-valued capacity with a nonempty finite carrier 𝑁, we will define the Choquet integral with respect to a discrete interval-valued capacity.

Definition 2.6. Let 𝑥∶𝑁→ℝ⁺ be a function such that 𝑥(𝑖)=𝑥_𝑖 for all 𝑖∈𝑁 and 𝜇 a discrete interval-valued capacity with a nonempty finite carrier 𝑁. The Choquet integral with respect to 𝜇 of 𝑥 is defined by 𝒞_𝜇𝑓=^𝑛−1_𝑖=0𝑥_𝜋(𝑖+1)−𝑥_𝜋(𝑖)𝜇𝐴_𝜋(𝑖+1), (2.2) where 𝜋 is a permutation on 𝑁 such that 𝑥_𝜋(0)=0≤𝑥_𝜋(1)≤𝑥_𝜋(2)≤⋯≤𝑥_𝜋(𝑛) and 𝐴_𝜋(𝑖)={𝜋(𝑖),…,𝜋(𝑛)} and 𝐴_𝜋(𝑛+1)=∅.

By (2.2), we can easily obtain the following basic property of 𝒞_𝜇.

Theorem 2.7. If 𝜇∈IVC_𝑁, then one has 𝒞_𝜇(𝑥)=𝒞_𝜇𝑙(𝑥),𝒞_𝜇𝑟(𝑥), (2.3) where a function 𝑥∶𝑁→ℝ⁺ by 𝑥(𝑖)=𝑥_𝑖 for all 𝑖∈𝑁.

From the right-hand side of (2.2), we note that ^𝑛_𝑖=0𝑥_𝜋(𝑖+1)−𝑥_𝜋(𝑖)𝜇𝐴_𝜋(𝑖+1)≠^𝑛_𝑖=1𝑥_𝜋(𝑖)𝜇𝐴_𝜋(𝑖+1)−𝜇𝐴_𝜋(𝑖), (2.4) in general. Because of this note, we consider a new difference operation ⊖ defined by[]⊖[]=[]𝑎,𝑏𝑐,𝑑𝑎−𝑐,𝑏−𝑑if0≤𝑎−𝑐≤𝑏−𝑑,0ifotherwise, (2.5) where 𝑎,𝑏,𝑐,𝑑∈ℝ⁺. From this difference operation, we can easily see that [𝑘𝑎,𝑘𝑏]⊖[𝑘𝑐,𝑘𝑑]=𝑘([𝑎,𝑏]⊖[𝑐,𝑑]) for all 𝑘,𝑎,𝑏,𝑐,𝑑∈ℝ⁺. We denote byIVC¹_𝑁(𝑥) the set of normalized interval-valued capacities 𝜇=[𝜇_𝑙,𝜇_𝑟] with a nonempty finite carrier 𝑁 satisfying the following condition: 𝜇_𝑙𝐴_𝜋(𝑖)−𝜇_𝑙𝐴_𝜋(𝑖+1)≤𝜇_𝑟𝐴_𝜋(𝑖)−𝜇_𝑟𝐴_𝜋(𝑖+1), (2.6) for all 𝑖∈𝑁 and where 𝜋 is a permutation on {1,2,…,𝑛} such that 𝑥_𝜋(1)≤𝑥_𝜋(2)≤⋯≤𝑥_𝜋(𝑛) and 𝐴_𝜋(𝑖)={𝜋(𝑖),…,𝜋(𝑛)} and 𝐴_𝜋(𝑛+1)=∅. We remark that IVC¹_𝑁(𝑥)⊂IVC_𝑁 and that if 𝜇∈IVC¹_𝑁(𝑥), then we have𝜇𝐴_𝜋(𝑖)⊖𝜇𝐴_𝜋(𝑖+1)=𝜇_𝑙𝐴_𝜋(𝑖)−𝜇_𝑙𝐴_𝜋(𝑖+1),𝜇_𝑟𝐴_𝜋(𝑖)−𝜇_𝑟𝐴_𝜋(𝑖+1). (2.7) By Theorem 2.7 and (1.2), (2.5), and (2.7), we derive the following theorem.

Theorem 2.8. If there exists function 𝑥∶𝑁→ℝ⁺ by 𝑥(𝑖)=𝑥_𝑖 for all 𝑖∈𝑁 and 𝜇∈IVC¹_𝑁(𝑥), then one has 𝒞_𝜇(𝑥)=^𝑛_𝑖=1𝑥_𝜋(𝑖)𝜇𝐴_𝜋(𝑖)⊖𝜇𝐴_𝜋(𝑖+1), (2.8) where 𝜋 is a permutation on 𝑁 such that 𝑥_𝜋(1)≤𝑥_𝜋(2)≤⋯≤𝑥_𝜋(𝑛) and 𝐴_𝜋(𝑖)={𝜋(𝑖),…,𝜋(𝑛)} and 𝐴_𝜋(𝑛+1)=∅.

Proof. By Theorem 2.7 and the definition (1.2) and the difference (2.5) operation, we have 𝒞_𝜇(𝑥)=𝒞_𝜇𝑙(𝑥),𝒞_𝜇𝑟(𝑥=_𝑛𝑖=1)𝑥_𝜋(𝑖)𝜇_𝑙𝐴_𝜋(𝑖)−𝜇_𝑙𝐴_𝜋(𝑖+1),^𝑛_𝑖=1𝑥_𝜋(𝑖)𝜇_𝑟𝐴_𝜋(𝑖)−𝜇_𝑟𝐴_𝜋(𝑖+1)=^𝑛_𝑖=1𝑥_𝜋(𝑖)𝜇_𝑙𝐴_𝜋(𝑖)−𝜇_𝑙𝐴_𝜋(𝑖+1),𝜇_𝑟𝐴_𝜋(𝑖)−𝜇_𝑟𝐴_𝜋(𝑖+1)=^𝑛_𝑖=1𝑥_𝜋(𝑖)𝜇_𝑙𝐴_𝜋(𝑖),𝜇_𝑟𝐴_𝜋(𝑖)⊖𝜇_l𝐴_𝜋(𝑖+1),𝜇_𝑟𝐴_𝜋(𝑖+1)=^𝑛_𝑖=1𝑥_𝜋(𝑖)𝜇𝐴_𝜋(𝑖)⊖𝜇𝐴_𝜋(𝑖+1). (2.9)

3. The Choquet Integral as an Interval-Valued Aggregation Operator

In this section, we define the interval-valued weighted arithmetic mean (IWAM), which is the concept of a generalized aggregation (or an uncertain aggregation), as follows:IWAM_𝑤(𝑥)=^𝑛_𝑖=1𝑤_𝑖𝑥_𝑖, (3.1) where 𝑤_𝑖=[𝑤_𝑙𝑖,𝑤_𝑟𝑖], 0≤𝑤_𝑙𝑖≤𝑤_𝑟𝑖≤1 for all 𝑖∈𝑁, ∑^𝑛_𝑖=1𝑤_𝑙𝑖≤∑^𝑛_𝑖=1𝑤_𝑟𝑖=1, and 𝑥∶𝑁→ℝ⁺ by 𝑥(𝑖)=𝑥_𝑖 for all 𝑖∈𝑁 is a function such that 𝑥₁,…,𝑥_𝑛 represent the arguments. We denote by 𝑥=(𝑥₁,…,𝑥_𝑛) an 𝑛-dimensional vector in (ℝ⁺)^𝑛.

Note that the arguments 𝑥₁,…,𝑥_𝑛 that are used in such an interval-valued aggregation process strongly depend upon the interval-valued (or uncertain) weight vector 𝑤=(𝑤₁,…,𝑤_𝑛)∈(𝐼([0,1]))^𝑛. Then the interval-valued Shannon-entropy 𝐻_𝑆 of 𝑤 defined on 𝑁 is given by𝐻_𝑆(𝑤)=^𝑛_𝑖=1ℎ𝑤_𝑖, (3.2) where ℎ(𝑤_𝑖)=[ℎ(𝑤_𝑙𝑖),ℎ(𝑤_𝑟𝑖)] for 0≤𝑤_𝑙𝑖≤𝑤_𝑟𝑖≤1 for all 𝑖∈𝑁, andℎ(𝑤)=−𝑤ln𝑤if0<𝑤≤1,0if𝑤=0, (3.3) for all 𝑤∈[0,1]. Then it means a measure of dispersion associated to the interval-valued weight vector of the interval-valued weighted arithmetic mean IWAM_𝑤. We also easily see that𝐻_𝑆(𝑤)=𝐻_𝑆𝑤_𝑙,𝐻_𝑆𝑤_𝑟, (3.4) where 𝑤=[𝑤_𝑙,𝑤_𝑟].

Now, we define the following interval-valued (or uncertain) ordered weight averaging (IOWA) operator.

Definition 3.1. Let 𝑤=(𝑤₁,…,𝑤_𝑛)∈(𝐼([0,1]))^𝑛 be an interval-valued weight vector such that 𝑤_𝑖=[𝑤_𝑙𝑖,𝑤_𝑟𝑖], 0≤𝑤_𝑙𝑖≤𝑤_𝑟𝑖≤1 for all 𝑖∈𝑁, and ∑^𝑛_𝑖=1𝑤_𝑙𝑖≤∑^𝑛_𝑖=1𝑤_𝑟𝑖=1. The interval-valued ordered weighted averaging (IOWA) operator on (ℝ⁺)^𝑛 is defined by IOWA_𝑤(𝑥)=^𝑛_𝑖=1𝑤_𝑖𝑥_𝜋(𝑖), (3.5) where 𝑥=(𝑥₁,…,𝑥_𝑛)∈(ℝ⁺)^𝑛 and 𝜋 is a permutation on 𝑁 such that 0≤𝑥_𝜋(1)≤𝑥_𝜋(2)≤⋯≤𝑥_𝜋(𝑛).

From Definition 3.1, we haveIOWA_𝑤(𝑥)=^𝑛_𝑖=1𝑤_𝑙𝑖𝑥_𝜋(𝑖),^𝑛_𝑖=1𝑤_𝑟𝑖𝑥_𝜋(𝑖). (3.6) By (3.5) and (3.6), we obtain that if 𝑠∈{𝑙,𝑟} and we write 𝑤_𝑠=(𝑤_𝑠1,…,𝑤_𝑠𝑛), then OWA_𝑤𝑠 is an ordered weighted averaging operator associated to a weight vector 𝑤_𝑠, proposed by Yager [18]. Remark that if 𝑤_𝑙𝑖=𝑤_𝑟𝑖=𝑤_𝑖 for all 𝑖∈𝑁 and we write 𝑤=(𝑤₁,…,𝑤_𝑛), then IWAM_𝑤(𝑥)=∑^𝑛_𝑖=1𝑤_𝑖𝑥_𝑖 is the weight arithmetic mean (WAM), 𝐻_𝑆(𝑤)=∑^𝑛_𝑖=1ℎ(𝑤_𝑖) is the Shannon entropy of 𝑤, and IOWA_𝑤(𝑥)=∑^𝑛_𝑖=1𝑤_𝑖𝑥_𝜋(𝑖) is the ordered weighted averaging (OWA) operator (see [3, 5]).

Theorem 3.2. (1) If one takes 𝑤_𝑙=𝑤_𝑟=(1/𝑛,…,1/𝑛), then 𝐻_𝑆(𝑤)=ln𝑛 is maximum.
(2) If one takes 𝑤_𝑙𝑖=1 and 𝑤_𝑟𝑗=1 for some 𝑖,𝑗∈𝑁, then 𝐻(𝑤)=0 is minimum.

Proof. (1) 𝐻_𝑆(𝑤)=[∑^𝑛_𝑖=1(−𝑤_𝑙1ln𝑤_𝑙𝑖),∑^𝑛_𝑖=1(−𝑤_𝑟𝑖ln𝑤_𝑟𝑖)]=[ln𝑛,ln𝑛]=ln𝑛.
(2) 𝐻_𝑆(𝑤)=[∑^𝑛_𝑘=1ℎ(𝑤_𝑙𝑘),∑^𝑛_𝑘=1ℎ(𝑤_𝑟𝑘)]=[ℎ(𝑤_𝑙𝑖),ℎ(𝑤_𝑟𝑗)]=[0,0]=0.

From Theorem 3.2, the interval-valued measure of dispersion can be normalized into1ln𝑛𝐻_𝑆(𝑤)=1ln𝑛^𝑛_𝑖=1ℎ𝑤_𝑖, (3.7) so that it ranges in 𝐼([0,1]). Finally, we will define the interval-valued entropy of an interval-valued capacity which is the generalization of the entropy proposed by Marichal [3] as follows:𝐻_𝑀𝜇=_{𝑖∈𝑁{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇(𝑇∪{𝑖})−𝜇(𝑇), (3.8) where 𝜇 is a capacity on 𝑁, |𝑇|=𝑡 is the cardinality of 𝑇, the coefficients 𝑝_𝑡(|𝑇|=𝑡) are nonnegative, and ∑_{𝑇⊂𝑁⧵{𝑖}}𝑝_𝑡=1.

Definition 3.3. The interval-valued entropy of an interval-valued capacity 𝜇 is defined by 𝐻_𝑀𝜇=_{𝑖∈𝑁𝑇⊂𝑁⧵{𝑖}}𝑝_𝑡ℎ𝜇(𝑇∪{𝑖})⊖𝜇(𝑇), (3.9) where ⊖ is the same operation in (2.5), the coefficients 𝑝_𝑡(|𝑇|=𝑡) are nonnegative, ∑_{𝑇⊂𝑁⧵{𝑖}}𝑝_𝑡=1, and ℎ is the same function in (3.3).

From Definition 3.3, if we take 𝜇_𝑙=𝜇_𝑟=𝜇, then 𝐻_𝑀(𝜇)=𝐻_𝑀(𝜇). We consider the following assumption of 𝜇=[𝜇_𝑙,𝜇_𝑟]:_{{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇_𝑙(𝑇∪{𝑖})−𝜇_𝑙(𝑇)≤_{{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇_𝑟(𝑇∪{𝑖})−𝜇_𝑟(𝑇),∀𝑖∈𝑁, (3.10) and let IVC²_𝑁={𝜇∈IVC_𝑁∣𝜇satisfyassumption(3.10)}.

Theorem 3.4. If 𝜇=[𝜇_𝑙,𝜇_𝑟]∈IVC²_𝑁, then one has 𝐻_𝑀𝜇=𝐻_𝑀𝜇_𝑙,𝐻_𝑀𝜇_𝑟. (3.11)

Proof. By Definition 3.3, we can directly calculate 𝐻_𝑀(𝜇) as follows: 𝐻_𝑀𝜇=_{𝑖∈𝑁{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇(𝑇∪{𝑖})⊖𝜇(𝑇=_{𝑖∈𝑁{𝑖}𝑇⊂𝑁⧵})𝑝_𝑡ℎ𝜇_𝑙(𝑇∪{𝑖})−𝜇_𝑙(𝑇),𝜇_𝑟(𝑇∪{𝑖})−𝜇_𝑟(𝑇)=_{𝑖∈𝑁{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇_𝑙(𝑇∪{𝑖})−𝜇_𝑙(𝑇),ℎ𝜇_𝑟(𝑇∪{𝑖})−𝜇_𝑟(𝑇)=_{𝑖∈𝑁{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇_𝑙(𝑇∪{𝑖})−𝜇_𝑙(𝑇),_{𝑖∈𝑁{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇_𝑟(𝑇∪{𝑖})−𝜇_𝑟(𝑇)=𝐻_𝑀𝜇_𝑙,𝐻_𝑀𝜇_𝑟. (3.12)

From Theorem 3.4, we can see that if we take 𝜇=[𝜇_𝑙,𝜇_𝑟]∈IVC_𝑁 such that 𝐻_𝑀(𝜇_𝑟)<𝐻_𝑀(𝜇_𝑙), then 𝐻_𝑀(𝜇) is not defined. Thus, the Assumption (3.10) of 𝜇 is a sufficient condition for defining the interval-valued entropy 𝐻_𝑀(𝜇) of an interval-valued capacity 𝜇. We also suggest that that 𝐻_𝑀(𝜇) is interpreted as an interval-valued measure of dispersion for 𝒞_𝜇 a sum over 𝑖∈𝑁 of an average value of ℎ(𝜇(𝑇∪{𝑖})⊖𝜇(𝑇))(𝑇⊂𝑁⧵{𝑖}) as follows: for all 𝑖∈𝑁,𝜇𝐴_𝜋(𝑖)⊖𝜇𝐴_𝜋(𝑖+1)=_{{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇(𝑇∪{𝑖})⊖𝜇(𝑇). (3.13)

Theorem 3.5. If 𝑥=(𝑥₁,…,𝑥_𝑛) and 𝜇∈IVC¹_𝑁(𝑥), then 𝜇∈IVC²_𝑁.

Proof. Let 𝑥=(𝑥₁,…,𝑥_𝑛)∈(ℝ⁺)^𝑛 and 𝜋 be a permutation on 𝑁 such that 0≤𝑥_𝜋(1)≤𝑥_𝜋(2)≤⋯≤𝑥_𝜋(𝑛). Since 𝜇∈IVC¹_𝑁(𝑥), we get 𝜇_𝑙(𝐴_𝜋(𝑖))−𝜇_𝑙(𝐴_𝜋(𝑖+1))≤𝜇_𝑟(𝐴_𝜋(𝑖))−𝜇_𝑟(𝐴_𝜋(𝑖+1)) for all 𝑖∈𝑁. Thus, by (3.13), _{{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇_𝑙(𝑇∪{𝑖})−𝜇_𝑙(𝑇)=𝜇_𝑙𝐴_𝜋(𝑖)−𝜇_𝑙𝐴_𝜋(𝑖+1)≤𝜇_𝑟𝐴_𝜋(𝑖)−𝜇_𝑟𝐴_𝜋(𝑖+1)=_{{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇_𝑟(𝑇∪{𝑖})−𝜇_𝑟(𝑇). (3.14) This implies 𝐻_𝑀(𝜇_𝑙)≤𝐻_𝑀(𝜇_𝑟). Therefore 𝜇∈IVC²_𝑁.

Theorem 3.6. If 𝑥=(𝑥₁,…,𝑥_𝑛), 𝜇∈IVC¹_𝑁(𝑥), and 𝒞_𝜇=IOWA_𝑤, then one has 𝐻_𝑀(𝜇)=𝐻_𝑆(𝑤), that is, 𝐻_𝑀𝜇=^𝑛_𝑖=1ℎ𝜇𝐴_𝜋(𝑖)⊖𝜇𝐴_𝜋(𝑖+1), (3.15) where ⊖ is the same operation in (2.3), ℎ is the same function in (2.8), and 𝑥=(𝑥₁,…,𝑥_𝑛)∈(ℝ⁺)^𝑛, and 𝜋 is a permutation on 𝑁 such that 0≤𝑥_𝜋(1)≤𝑥_𝜋(2)≤⋯≤𝑥_𝜋(𝑛)

Proof. Since 𝒞_𝜇=IOWA_𝑤, 𝑤_𝑖=𝜇(𝐴_𝜋(𝑖))⊖𝜇(𝐴_𝜋(𝑖+1)). By (3.13), we get 𝐻_𝑀𝜇=_{𝑖∈𝑁{𝑖}𝑇⊂𝑁⧵}𝑝_𝑡ℎ𝜇(𝑇∪{𝑖})⊖𝜇(𝑇=_𝑛𝑖=1)ℎ𝜇𝐴_𝜋(𝑖)⊖𝜇𝐴_𝜋(𝑖+1)=^𝑛_𝑖=1ℎ(𝑤)=𝐻_𝑆(𝑤). (3.16)

4. Applications

In this section, we consider the problem of evaluating students in a high school with respect to three subjects: mathematics (M), physics (P), and literature (L), proposed by Marichal [3]. Suppose that the school is more scientifically than literary from somewhat oriented to extremely oriented, so that interval-valued weights could be, for example, 𝑤_M=[1/4,1/2], 𝑤_P=[1/4,3/8], and 𝑤_L=[1/8,1/8], respectively. We note that 𝑤=(𝑤_M,𝑤_P,𝑤_L) and IWAM_𝑤(𝑥)=[WAM_𝑤𝑙(𝑥),WAM_𝑤𝑟(𝑥)] for all 𝑥∈{𝑎,𝑏,𝑐}.

If we take 𝑤=[𝑤_𝑙,𝑤_𝑟], then 𝑤_𝑙=(1/4,1/4,1/8) and 𝑤_𝑟=(1/2,3/8,1/8). Then the interval-valued weighted arithmetic mean will give the results for three students 𝑎,𝑏, and 𝑐 (marks are given on a scale from 0 to 20) (see Table 1).

Table 1 

We note that IWAM_𝑤(𝑎)>IWAM_𝑤(𝑐)>IWAM_𝑤(𝑏). The total interval-valued weight is from rather well distributed to quite well distributed over three subjects since we have 1ln𝑛𝐻_𝑆(𝑤)=[0.868,0.887]. (4.1)

We consider the 𝛼-mean evaluation 𝐸_𝛼 of IWAM_𝑤𝑙 as follows:𝐸_𝛼(𝑥)=𝛼WAM_𝑤𝑙(𝑥)+(1−𝛼)WAM_𝑤𝑟(𝑥) (4.2) for all 𝛼∈[0,1] and 𝑥∈{𝑎,𝑏,𝑐}. The 𝛼-mean evaluation 𝐸_𝛼 implies that we can interpret the difference of the degree of favor for students. Indeed, if 𝛼=0, that is, if the school is more scientifically than literary extremely oriented, then the school wants to favor more student 𝑐 as 𝐸₀(𝑎)−𝐸₀(𝑐)=1.75; if 𝛼=1, that is, if the school is more scientifically than literary somewhat oriented, then the school wants to favor more student 𝑐 as 𝐸₁(𝑎)−𝐸₁(𝑐)=0.575.

Now, if the school wants to favor somewhat well-equilibrated over extremely well equilibrated students without weak points, then student 𝑐 should be considered better than student 𝑎, who has a severe weakness in literature. Unfortunately, no interval-valued vector (𝑤_M,𝑤_P,𝑤_L) satisfying 𝑤_M>𝑤_P>𝑤_L is able to favor student 𝑐. Indeed, it is possible that IWAM_𝑤(14,15,15)>IWAM_𝑤(18,16,10)⟺𝑤_L>𝑤_M. (4.3)

The reason of this problem is that much importance is given to mathematics and physics, which present some overlap effect since, usually, students from little good to rather good at mathematics are also from little good to rather good at physics (and vice versa), so that the interval-valued evaluation is overestimated (resp., underestimated) for students from little good to rather good (resp., from little bad to rather bad) at mathematics and/or physics.

This problem can be overcome by using a suitable interval-valued capacity 𝜇 and the Choquet integral 𝒞_𝜇 as follows.

(i) Since scientific subjects are more important than literature, the following interval-valued weights can be put on subjects taken individually: 𝜇({M})=[0.25,0.5], 𝜇({P})=[0.25,0.375] and 𝜇({L})=[0.125,0.125]. Note that the initial interval-valued ratio of interval-valued weight ([2,4],[2,3],[1,1]) is kept unchanged.

(ii) Since mathematics and physics overlap, the interval-valued weight attributed to the pair {M,P} should be less than the sum of the interval-valued weight of mathematics and physics: 𝜇({M,P})=[0.3,0.6].

(iii) Since students equally good at scientific subjects and literature must be favored, the interval-valued weight attributed to the pair {L,M} should be greater than the sum of individual interval-valued weights (the same for physics and literature): 𝜇({M,L})=[0.45,0.75]=𝜇({P,L}).

(iv) 𝜇(∅)=0 and 𝜇({M,P,L})=[0.55,1].

If we take 𝜇=[𝜇_𝑙,𝜇_𝑟], and 𝜇_𝑙({L,P,M})=0.55, 𝜇_𝑟({L,P,M})=1, 𝜇_𝑙({P,M})=0.3, 𝜇_𝑟({P,M})=0.6, 𝜇_𝑙({M,L})=0.45, 𝜇_𝑟({M,L})=0.75, 𝜇_𝑙({P,L})=0.45, 𝜇_𝑟({P,L})=0.75, 𝜇_𝑙({P})=0.25, 𝜇_𝑟({P})=0.375, 𝜇_𝑙({M})=0.25, 𝜇_𝑟({M})=0.5, 𝜇_𝑙({L})=0.125, and 𝜇_𝑟({L})=0.125, then 𝐻_M(𝜇_𝑙)=0.8802<0.9974=𝐻_M(𝜇_𝑟) and hence 𝜇∈IVC²_𝑁.

Applying the Choquet integral with respect to the above interval-valued capacity leads to the Choquet integrals see Table 2.

Table 2 

Since IWAM_𝑤(𝑎)>IWAM_𝑤(𝑐)>IWAM_𝑤(𝑏) and 𝒞_𝜇(𝑐)>𝒞_𝜇(𝑎)>𝒞_𝜇(𝑏), we can see that if we use IWAM_𝑤, then student 𝑎 has the best rank, but if we use 𝒞_𝜇, then student 𝑐 has the best rank. We also consider the 𝛼-mean Choquet evaluation 𝐸^𝑐_𝛼 of 𝒞_𝜇 as follows:𝐸^𝑐_𝛼(𝑥)=𝛼𝒞_𝜇𝑙(𝑥)+(1−𝛼)𝒞_𝜇𝑟(𝑥) (4.4) for all 𝛼∈[0,1] and 𝑥∈{𝑎,𝑏,𝑐}.

The 𝛼-mean Choquet evaluation 𝐸^𝑐_𝛼 implies that we can interpret the difference of the degree of favor for student 𝑎 and student 𝑐. Indeed, if 𝛼=1, that is, if the school wants to favor extremely well-equilibrated students, then the school wants to favor student 𝑐 than student 𝑎 as 𝐸^𝑐₁(𝑐)−𝐸^𝑐₁(𝑎)=0.35; if 𝛼=0, that is, if the school wants to favor somewhat more well-equilibrated students, then the school wants to favor student 𝑐 more than student 𝑎 as 𝐸^𝑐₀(𝑐)−𝐸^𝑐₀(𝑎)=0.15.

Finally, we have the normalized entropy of interval-valued capacity in IVC²_𝑁 as follows: 1ln𝑛𝐻_M𝜇=1𝐻ln𝑛^M𝜇_𝑙,1𝐻ln𝑛^M𝜇_𝑟,𝐻_M𝜇_𝑙=_{{𝑇⊂𝑁⧵P}}(3−𝑡−1)!𝑡!ℎ𝜇3!^𝑙(𝑇∪{P})−𝜇_𝑙(𝑇)+_{{𝑇⊂𝑁⧵M}}(3−𝑡−1)!𝑡!ℎ𝜇3!^𝑙(𝑇∪{M})−𝜇_𝑙(𝑇)+_{{𝑇⊂𝑁⧵L}}(3−𝑡−1)!𝑡!ℎ𝜇3!^𝑙(𝑇∪{L})−𝜇_𝑙(𝑇),𝐻_M𝜇_𝑙=_{{𝑇⊂𝑁⧵P}}(3−𝑡−1)!𝑡!ℎ𝜇3!^𝑟(𝑇∪{P})−𝜇_𝑟(𝑇)+_{{𝑇⊂𝑁⧵M}}(3−𝑡−1)!𝑡!ℎ𝜇3!^𝑟(𝑇∪{M})−𝜇_𝑟(𝑇)+_{{𝑇⊂𝑁⧵L}}(3−𝑡−1)!𝑡!ℎ𝜇3!^𝑟(𝑇∪{L})−𝜇_𝑟(𝑇), (4.5) where 𝑁={P,M,L}. Thus, we have 1ln𝑛𝐻_M𝜇=[0.8012,0.9079], (4.6) which shows that the total interval-valued weight is from still rather well distributed to very quite well distributed.

5. Conclusions

In this paper we consider the new interval-valued measure of the entropy of an interval-valued capacity which generalizes a measure of the entropy proposed by Marichal's [3]. From (3.1), (3.5), and (3.9) and Theorems 3.4, 3.5, 3.6, we investigate the interval-valued weighted arithmetic mean and interval-valued ordered weighted averaging operator for representing uncertain weight vectors which are used in the concept of an uncertain aggregations.

From an example in Section 4, it is possible that we use from somewhat oriented to extremely oriented instead of oriented, from rather well distributed to quite well distributed instead of well distributed, and from somewhat well equilibrated to extremely well equilibrated instead of well equilibrated in the problem of evaluating students.

In the future, by using these results of this paper, we can develop various problems and models for representing uncertain weights related to interacting criteria.

Acknowledgment

This paper was supported by the Konkuk University in 2012.