Constructing of fuzzy integer valued mapping and the application of its integral

Abstract

In this paper, based on statistical method, we establish a method for constructing fold line type fuzzy integer valued mapping to represent the change rule of a uncertain integer quantity which changes with the change of some quantity; For the convenience of calculations, we provide the calculation formula (in cut set) of linear operations of trapezoid-type fuzzy integers; And we introduce the concept of integral of fuzzy integer valued mapping by using the definition of integral of continuous fuzzy number valued mapping, investigate its properties, and obtain its calculating formula in cut set; At last, taking the case of estimating the quantity of daily vehicle flow of a road, we propose a method for obtaining some imprecise integer statistical quantities based on the theory set up by us, and using a practical example to show the specific algorithm of this method.

Keywords

Fuzzy number fuzzy integer fuzzy integer valued mapping constructing integral

1 Introduction

Statistical theory and method are the basic methods used in the field of modern engineering and social applications. With the development of mathematics, science and computer, the application of statistical methods is more and more extensive. For instance, in the statistical or experimental results, if the information intended to obtain can not be represented by using accurate digital data, we can consider using statistical methods to construct fuzzy numbers to represent these information (for example, see [12 , 19]).

Continuous fuzzy numbers and discrete fuzzy numbers are all special fuzzy sets. The concept of continuous fuzzy numbers and the concept of discrete fuzzy numbers are proposed by Chang and Zadeh in 1972 in [4] and by Voxman in 2001 in [20], respectively. With the development of theories and applications of fuzzy numbers (continuous fuzzy numbers and discrete fuzzy numbers), the concepts becomes more and more important. Recently, there is still a lot of work about continuous fuzzy numbers. For example, in [9], Nan, Zhang and Li developed a methodology for solving matrix games with payoffs of trapezoidal intuitionistic fuzzy numbers; In [1], Arotaritei and Ionescu introduced ařfuzzy Voronoiaś diagrams for fuzzy numbers of dimension two by extension of Voronoi diagrams for fuzzy numbers; In [5], Coroianu and Gagolewski studied the problem of the nearest approximation of fuzzy numbers by piecewise linear 1-knot fuzzy numbers. In [21], Yeh studied the existence of interval, triangular, and trapezoidal approximations of fuzzy numbers under a general condition; In [7], Huang and Wu et al constructed some approximations comprising fuzzy number sequences with useful properties for a general fuzzy number by using the convolution method. In [13], Wang and Li proposed an approximation of fuzzy number membership functions by using simple fuzzy number membership functions.

Likewise, about discrete fuzzy numbers, there are also many research results. In [2, 3], Casasnovas and Riera studied some characters of discrete fuzzy numbers. In 2013, Riera and Torrens studied the residual implications on the set of discrete fuzzy numbers in [10]. In 2014, Riera and Torrens introduced aggregation functions on the set of discrete fuzzy numbers whose support is a set of consecutive natural numbers from a couple of discrete aggregation functions in [11]. Massanet and Riera et al set up a new linguistic computational model based on discrete fuzzy numbers for computing with words in [8]. Recently, in [17], Wang and Shi et al defined two-dimensional discrete fuzzy numbers, gave a method for constructing two-dimensional fuzzy integer, and applied it to two factor evaluation in the uncertain environment. In [15 , 19], Wang, Nan et al studied fuzzy integers, and set up the methods for constructing fuzzy integer to represent imprecise integer quantity information.

In this paper, we study the problem of constructing fold line type fuzzy integer valued mapping, define the integral of fuzzy integer valued mapping, and propose a method for obtaining some imprecise integer statistical quantities based on the theory set up by us. Specific arrangements are as follows: In Section 2, we briefly review some basic notions, definitions and results about fuzzy numbers (continuous and discrete). In Section 3, On the basis of the methods of constructing fuzzy integer, we establish a method for constructing fold line type fuzzy integer valued mapping to represent the change rule of a uncertain integer quantity which changes with the change of some quantity by using statistical method. In Section 4, for the convenience of calculations in applications, we provide the calculation formula (in cut set) of linear operations of trapezoid-type fuzzy integers. In Section 5, we give a definition of integral of fuzzy integer valued mapping based on the definition of integral of continuous fuzzy number valued mapping, investigate its properties, and obtain its calculating formula in cut set. In Section 6, taking the case of estimating the quantity of daily vehicle flow of a road, we propose a method for obtaining some imprecise integer statistical quantities based on the theory set up by us, and give a practical example to show the specific algorithm of this method. In section 7, we make a summary of this paper.

2 Basic definition and notation

Let R be the real number set, and let I be the integer set. And let K (R) denote the collection of non-empty compact subsets of R. The addition, scalar multiplication and multiplication on the space K (R) are respectively defined as A + B = {a + b | a ∈ A, b ∈ B} and λA = {λa | a ∈ A} for any A, B ∈ K (R) , λ ∈ R.

A fuzzy subset (for short, a fuzzy set) of R is a function u : R → [0, 1]. For fuzzy set u, let [u] ^r = {x ∈ R : u (x) ≥ r} for any r ∈ (0, 1] be its r- level set. By supp (u) we denote the support of u, i.e., the set {x ∈ R : u (x) >0}. And we denote the closure of the supp (u) by [u] ⁰, i.e., $[u]^{0} = \bar{{x \in R : u (x) > 0}}$ .

For any fuzzy sets u, v and real number k, we define the addition and the multiplication of u and v, and the scalar multiplication of k and u by the following: $\begin{matrix} (u + v) (x) & = & sup_{y + z = x} min {u (y), v (z)} \\ (ku) (x) & = & {\begin{matrix} u (x / k) & if k \neq 0 \\ \hat{0} (x) & if k = 0 \end{matrix} \end{matrix}$ where $\hat{0} (x) = {\begin{matrix} 1 & if x = 0 \\ 0 & if x \neq 0 \end{matrix}$ (for any a ∈ R, $\hat{a}$ can be similarly defined).

If u is a normal and fuzzy convex fuzzy set of R, u (x) is upper semi-continuous, and [u] ⁰ is compact, then we call u a (continuous) fuzzy number, and denote the collection of all fuzzy numbers by E.

It is known that if u ∈ E, then for each r ∈ [0, 1], [u] ^r is a closed interval (denoted as $[\underline{u} (r), \bar{u} (r)]$ ) in R.

Let a₀, a₁, b₁ and b₀ ∈ R with a₀ ≤ a₁ ≤ b₁ ≤ b₀. If the fuzzy set u : R → [0, 1] is defined as $u (x) = {\begin{matrix} \frac{x - a_{0}}{a_{1} - a_{0}} & if & x \in [\underline{c}, a_{1}) \\ 1 & if & x \in [a_{1}, b_{1}] \\ \frac{b_{0} - x}{b_{0} - b_{1}} & if & x \in (b_{1}, \bar{c}] \\ 0 & if & x \notin [\underline{c}, \bar{c}] \end{matrix}$ then we call u a trapezoid-type fuzzy number, and denote it as $u = (a_{0} |_{\underline{c}}, a_{1}, b_{1}, b_{0} |_{\bar{c}})$ . Specially, if a₁ = b₁ (denoted d), then we call the trapezoid-type fuzzy number u a triangle-type fuzzy number, and denote it as $u = (a_{0} |_{\underline{c}}, d, b_{0} |_{\bar{c}})$ , where $\underline{c}, \bar{c} \in R$ with $a_{0} \leq \underline{c} \leq a_{1}$ and $b_{1} \leq \bar{c} \leq b_{0}$ . And we denote the collection of all trapezoid-type fuzzy numbers by Tra - E, and the collection of all triangle-type fuzzy numbers by Tri - E.

If $\underline{c} = a_{0}$ and $\bar{c} = b_{0}$ , then trapezoid-type fuzzy number $(a_{0} |_{\underline{c}}, a_{1}, b_{1}, b_{0} |_{\bar{c}})$ and triangle-type fuzzy number $(a_{0} |_{\underline{c}}, d, b_{0} |_{\bar{c}})$ are called trapezoid fuzzy number (denoted as (a₀, a₁, b₁, b₀)) and triangle fuzzy number (denoted as (a₀, d, b₀)),respectively.

For any s₁, s₂ ∈ I with s₁ ≤ s₂, we denote 〈s₁, s₂〉 = {x ∈ I : s₁ ≤ x ≤ s₂} and call it a closed integer interval.

A fuzzy set u : R → [0, 1] is called a fuzzy integer [15] if its support is a closed integer interval (denoted as $〈 \underline{u} (0), \bar{u} (0) 〉$ ), and satisfies: (1) u is normal, i.e., there exists an $\hat{x} \in 〈 \underline{u} (0), \bar{u} (0) 〉$ such that $u (\hat{x}) = 1$ ; (2) u (x_i) ≤ u (x_j) for any $x_{i}, x_{j} \in 〈 \underline{u} (0), \hat{x} 〉$ with x_i ≤ x_j; (3) u (x_i) ≥ u (x_j) for any $x_{i}, x_{j} \in 〈 \hat{x}, \bar{u} (0) 〉$ with x_i ≤ x_j. And we denote the collection of all fuzzy integers by F_I.

If u ∈ F_I, then for each r ∈ [0, 1], [u] ^r is a closed integer interval. For u ∈ F_I, we denote the closed integer interval [15] as $[u]^{r} = 〈 \underline{u} (r), \bar{u} (r) 〉$ .

For any u, v ∈ F_I, we have that u + v ∈ F_I and $〈 \underline{u + v} (r), \bar{u + v} (r) 〉 = 〈 \underline{u} (r) + \underline{v} (r), \bar{u} (r) + \underline{v} (r) 〉$ for any r ∈ [0, 1] (see Theorem 3.4 in [15]).

For any x ∈ R, we use ⌊x⌉ to indicate the integer which is obtained by arithmetic rounding to x.

For u ∈ F_I, k ∈ R, we have that k ∘ u ∈ F_I, and $\underline{k \circ u} (r) = ⌊ k \underline{u} (r) ⌉$ and $\bar{k \circ u} (r) = ⌊ k \bar{u} (r) ⌉$ as k ≥ 0, $\underline{k \circ u} (r) = ⌊ k \bar{u} (r) ⌉$ and $\bar{k \circ u} (r) = ⌊ k \underline{u} (r) ⌉$ as k < 0 (see Theorem 3.6 in [15], and see Definition 3.2 in [15] about the definition of the scalar multiplication “∘”).

For any u, v ∈ F_I, if there exists w ∈ F_I such that u = v + w, then we say u, v to be havingH-difference, and call w the H-difference of u and v, denoted as u - v.

Let s₀, s₁, t₁ and t₀ ∈ I with s₀ ≤ s₁ ≤ t₁ ≤ t₀. If the fuzzy set u : R → [0, 1] is defined as $u (x) = {\begin{matrix} \frac{x - s_{0}}{s_{1} - s_{0}} & if & x \in 〈 \underline{m}, s_{1} 〉 \\ 1 & if & x \in 〈 s_{1}, t_{1} 〉 \\ \frac{t_{0} - x}{t_{0} - t_{1}} & if & x \in 〈 t_{1}, \bar{m} 〉 \\ 0 & if & x \notin 〈 \underline{m}, \bar{m} 〉 \end{matrix}$ then we call (see Theorem 4.1 and Definition 4.1 in [15]) u a trapezoid-type fuzzy integer, and denote it as $u = F_{I} (s_{0} |_{\underline{m}}, s_{1}, t_{1}, t_{0} |_{\bar{m}})$ . Specially, if s₁ = t₁ (denoted n), then we call the trapezoid-type fuzzy integer u a triangle-type fuzzy integer, and denote it as $u = F_{I} (s_{0} |_{\underline{m}}, n, t_{0} |_{\bar{m}})$ , where $\underline{m}, \bar{m} \in I$ with $s_{0} \leq \underline{m} \leq s_{1}$ and $t_{1} \leq \bar{m} \leq t_{0}$ .

And we denote the collection of all trapezoid-type fuzzy integers by Tra - F_I, and the collection of all triangle-type fuzzy integers by Tri - F_I.

Remark 1. It is obvious that as $\underline{m} = s_{0}$ and $\bar{m} = t_{0}$ , trapezoid-type fuzzy integer $F_{I} (s_{0} |_{\underline{m}}, s_{1}, t_{1}, t_{0} |_{\bar{m}})$ and triangle-type fuzzy integer $F_{I} (s_{0} |_{\underline{m}}, n, t_{0} |_{\bar{m}})$ become trapezoid fuzzy integer F_I (s₀, s₁, t₁, t₀) and triangle fuzzy integer F_I (s₀, n, t₀) (see Proposition 3.4 and Definition 3.1 in [18]), respectively.

3 Constructing of fuzzy integer valued mapping

We know that fuzzy integer number can bee used to express uncertain or imprecise integer quantity like the number of animal or plant in a certain area, such as the people’s number of “a swarm of people”, the trees’ number of “A large forest” and so on. Likewise, fuzzy integer valued mapping can be used to represent uncertain or imprecise integer variable quantity that changes with the change of some factor (quantity), such as the approximate variation in the quantity of animals that live a year in a certain area, the about number of vehicles passing through a road at any moment t and so on. In [15], Wang, Nan and Li given the methods of constructing fuzzy integer numbers to represent uncertain or imprecise integer information. In this section, we establish methods of constructing fuzzy integer valued mapping to represent uncertain or imprecise integer variable quantity that change with the change of some factor(quantity).

Problem: Consider a uncertain or imprecise integer variable quantity (denoted by u) which changes with the change of some certain and precise variable quantity (denoted by t). Thus, a fuzzy integer valued mapping is defined: F : D → F_I, t → F (t) = u, ∀ t ∈ D, where D is the range of the certain and precise variable quantity t, ie., the domain of the mapping F. Suppose the following data about the uncertain or imprecise integer variable quantity u are respectively obtained by statistical methods as t is at positions t₁, t₂, ⋯ , t_m, i.e., t = t₁, t₂, ⋯ , t_m with t₁ < t₂ < ⋯ < t_m: $\begin{matrix} u \\ t = t_{1} & a_{11} & a_{12} & \dots & a_{1 n_{1}} \\ t = t_{2} & a_{21} & a_{22} & \dots & a_{2 n_{2}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ t = t_{m} & a_{m 1} & a_{m 2} & \dots & a_{{mn}_{m}} \end{matrix}$ (1) where a_ij ∈ I is the jth statistic value of u as t = t_i (i = 1, 2, ⋯ , m, j = 1, 2, ⋯ n_i).

In the following, from the statistic data equation (1), we establish a fold line type fuzzy integer valued mapping to approximate the fuzzy integer valued mapping F.

The constructing method:

First: For each i = 1, 2, ⋯ , n, we work out the mean μ_i of the statistic values a_i1, a_i2, ⋯ , a_{in
_i} of the uncertain or imprecise integer variable quantity u as t = t_i: $μ_{i} = \frac{1}{n_{i}} \sum_{j = 1}^{n_{i}} a_{ij}$ (2)

Second: For each i = 1, 2, ⋯ , n, we work out the left separation degrees Lσ_i and the right separation degrees Rσ_i of the statistic values a_i1, a_i2, ⋯ , a_{in
_i} of the uncertain or imprecise integer variable quantity u as t = t_i, respectively:

$\begin{matrix} L σ_{i} & = & \frac{1}{N_{Li}} \sum_{a_{ij} < μ_{i}} (μ_{i} - a_{ij}) \\ R σ_{i} & = & \frac{1}{N_{R_{i}}} \sum_{a_{ij} > μ_{i}} (a_{ij} - μ_{i}) \end{matrix}$ (3) where N_{L
_i} and N_{R
_i} are the number of the statistic values which satisfy a_ij ≤ μ_i (j = 1, 2, ⋯ , n_i) in a_i1, a_i2, ⋯ , a_{in
_i}, and the number of the statistic values which satisfy a_ij ≥ μ_i (j = 1, 2, ⋯ , n_i) in a_i1, a_i2, ⋯ , a_{in
_i}, respectively.

Third: Make a domain 〈α, β〉 (such that the all possible statistic values of the uncertain or imprecise integer variable quantity u are in it) of the statistic value of the uncertain or imprecise integer variable quantity u according to the practical case. For each i = 1, 2, ⋯ , n, denote $(max_{_})_{i} = max {\underline{μ_{i} - λ L σ_{i}}, α}$ and $(min^{¯})_{i} = min {\bar{μ_{i} + λ R σ_{i}}, β}$ , where $\underline{μ_{i} - λ L σ_{i}} = max {n \in I : n \leq μ_{i} - λ L σ_{i}}$ , $\bar{μ_{i} + λ R σ_{i}} = min {n \in I : n \geq μ_{i} + λ R σ_{i}}$ , and λ is a parameter, that may be chosen in interval [2, 4] according to practical case.

(1) When the uncertain or imprecise integer variable quantity u is of Two-sided type (see Fig. 1 in [15]), for each i = 1, 2, ⋯ , n, we construct a trapezoid-type fuzzy integer u_i as

$\begin{matrix} u_{i} \\ = F_{I} (\underline{μ_{i} - λ L σ_{i}} |_{(max_{_})_{i}}, {\underline{N}}_{i}, {\bar{N}}_{i}, \bar{μ_{i} + λ R σ_{i}} |_{(min^{¯})_{i}}) \end{matrix}$ (4) where $\underline{N} = max {n \in I : n \leq μ_{i}}$ and $\bar{N} = min {n \in I : n \geq μ_{i}}$ .

(2) When the uncertain or imprecise integer variable quantity u is of Right-sided type (see Fig. 2 in [15]), for each i = 1, 2, ⋯ , n, we construct a trapezoid-type fuzzy integer u_i as $u_{i} = F_{I} (\underline{μ_{i} - λ L σ_{i}} |_{(max_{_})_{i}}, {\underline{N}}_{i}, β, β |_{β})$ (5) where ${\underline{N}}_{i} = max {n \in I : n \leq μ_{i}}$ .

(3) When the uncertain or imprecise integer variable quantity u is of Left-sided type (see Fig. 3 in [15]), for each i = 1, 2, ⋯ , n, we construct a trapezoid-type fuzzy integer u_i as $u_{i} = F_{I} (α |_{α}, α, {\bar{N}}_{i}, \bar{μ_{i} + λ R σ_{i}} |_{(min^{¯})_{i}})$ (6) where ${\bar{N}}_{i} = min {n \in I : n \geq μ_{i}}$ .

Fourth: We construct a fold line type fuzzy integer valued mapping $\hat{F}$ (i.e., $u = \hat{F} (t)$ ) to approximate the fuzzy integer valued mapping F as following:

$\begin{matrix} \hat{F} (t) \\ = {\begin{matrix} \frac{t_{2} - t}{t_{2} - t_{1}} \circ u_{1} + \frac{t - t_{1}}{t_{2} - t_{1}} \circ u_{2}, & t \in [t_{1}, t_{2}) \\ ⋮ \\ \frac{t_{i} - t}{t_{i} - t_{i - 1}} \circ u_{i - 1} + \frac{t - t_{i - 1}}{t_{i} - t_{i - 1}} \circ u_{i}, & t \in [t_{i - 1}, t_{i}) \\ ⋮ \\ \frac{t_{m} - t}{t_{m} - t_{m - 1}} \circ u_{m - 1} + \frac{t - t_{m - 1}}{t_{m} - t_{m - 1}} \circ u_{m}, & t \in [t_{m - 1}, t_{m}] \end{matrix} \end{matrix}$ (7)

4 The linear operations of trapezoid-type fuzzy integers

For the convenience of calculations, we study the linear operations of trapezoid-type fuzzy integers in the following:

For any a ∈ R, we denote ⌈a ⌉ = min {n ∈ I : n ≥ a} and ⌊a ⌋ = max {a ∈ I : n ≤ a}.

Theorem 1. Let u ∈ Tra - F_I (can be denoted as $u = F_{I} (\underline{u} (0) |_{\underline{m}}, \underline{u} (1), \bar{u} (1), \bar{u} (0) |_{\bar{m}})$ ) and k ∈ R. We have that as k ≥ 0 $\begin{matrix} \underline{k \circ u} (r) \\ = {\begin{matrix} ⌊ k \underline{m} ⌉, & r \in [0, u (\underline{m})] \\ ⌊ k ⌈ \underline{u} (0) + r (\underline{u} (1) - \underline{u} (0)) ⌉ ⌉, & r \in (u (\underline{m}), 1] \end{matrix} \\ \bar{k \circ u} (r) \\ = {\begin{matrix} ⌊ k \bar{m} ⌉, & r \in [0, u (\bar{m})] \\ ⌊ k ⌊ \bar{u} (0) - r (\bar{u} (0) - \bar{u} (1)) ⌋ ⌉, & r \in (u (\bar{m}), 1] \end{matrix} \end{matrix}$ and as k < 0, $\begin{matrix} \underline{k \circ u} (r) \\ = {\begin{matrix} ⌊ k \bar{m} ⌉, & r \in [0, u (\bar{m})] \\ ⌊ k ⌊ \bar{u} (0) - r (\bar{u} (0) - \bar{u} (1)) ⌋ ⌉, & r \in (u (\bar{m}), 1] \end{matrix} \\ \bar{k \circ u} (r) \\ = {\begin{matrix} ⌊ k \underline{m} ⌉, & r \in [0, u (\underline{m})] \\ ⌊ k ⌈ \underline{u} (0) + r (\underline{u} (1) - \underline{u} (0)) ⌉ ⌉, & r \in (u (\underline{m}), 1] \end{matrix} \end{matrix}$

Proof. From $u = F_{I} (\underline{u} (0) |_{\underline{m}}, \underline{u} (1), \bar{u} (1), \bar{u} (0) |_{\bar{m}})$ , we see

$\underline{u} (r) = {\begin{matrix} \underline{m} & if & r \in [0, u (\underline{m})] \\ ⌈ \underline{u} (0) + r (\underline{u} (1) - \underline{u} (0)) ⌉ & if & r \in (u (\underline{m}), 1] \end{matrix}$ and

$\bar{u} (r) = {\begin{matrix} \bar{m} & if & r \in [0, u (\bar{m})] \\ ⌊ \bar{u} (0) - r (\bar{u} (0) - \bar{u} (1)) ⌋ & if & r \in (u (\bar{m}), 1] \end{matrix}$

Therefore, by $\underline{k \circ u} (r) = ⌊ k \underline{u} (r) ⌉$ , $\bar{k \circ u} (r) = ⌊ k \bar{u} (r) ⌉$ as k ≥ 0, and $\underline{k \circ u} (r) = ⌊ k \bar{u} (r) ⌉$ , $\bar{k \circ u} (r) = ⌊ k \underline{u} (r) ⌉$ as k < 0 for any r ∈ [0, 1] (Theorem 3.6 in [15]), we can obtain the the conclusion of the theorem.

Theorem 2. Let u, v ∈ Tra - F_I (can be denoted as $u = F_{I} (\underline{u} (0) |_{\underline{m}}, \underline{u} (1), \bar{u} (1), \bar{u} (0) |_{\bar{m}})$ and $v = F_{I} (\underline{v} (0) |_{\underline{n}}, \underline{v} (1), \bar{v} (1), \bar{v} (0) |_{\bar{n}})$ ). We have that

$\begin{matrix} \underline{u + v} (r) \\ = {\begin{matrix} \underline{m} + \underline{n}, & r \in [0, u (\underline{m})] \\ ⌈ \underline{u} (0) + r (\underline{u} (1) - \underline{u} (0)) ⌉ + \underline{n}, & r \in (u (\underline{m}), v (\underline{n})] \\ \begin{matrix} ⌈ \underline{u} (0) + r (\underline{u} (1) - \underline{u} (0)) ⌉ \\ + ⌈ \underline{v} (0) + r (\underline{v} (1) - \underline{v} (0)) ⌉ \end{matrix}, & r \in (v (\underline{n}), 1] \end{matrix} \end{matrix}$ (8) as $u (\underline{m}) \leq v (\underline{n})$ ;

$\begin{matrix} \underline{u + v} (r) \\ = {\begin{matrix} \underline{m} + \underline{n}, & r \in [0, v (\underline{n})] \\ \underline{m} + ⌈ \underline{v} (0) + r (\underline{v} (1) - \underline{v} (0)) ⌉, & r \in (v (\underline{n}), u (\underline{m})] \\ \begin{matrix} ⌈ \underline{u} (0) + r (\underline{u} (1) - \underline{u} (0)) ⌉ \\ + ⌈ \underline{v} (0) + r (\underline{v} (1) - \underline{v} (0)) ⌉ \end{matrix}, & r \in (u (\underline{m}), 1] \end{matrix} \end{matrix}$ (9)

as $u (\underline{m}) > v (\underline{n})$ , and

$\begin{matrix} \bar{u + v} (r) \\ = {\begin{matrix} \bar{m} + \bar{n}, & r \in [0, u (\bar{m})] \\ ⌊ \bar{u} (0) - r (\bar{u} (0) - \bar{u} (1)) ⌋ + \bar{n}, & r \in (u (\bar{m}), v (\bar{n})] \\ \begin{matrix} ⌊ \bar{u} (0) - r (\bar{u} (0) - \bar{u} (1)) ⌋ \\ + ⌊ \bar{v} (0) - r (\bar{v} (0) - \bar{v} (1)) ⌋ \end{matrix}, & r \in (v (\bar{n}), 1] \end{matrix} \end{matrix}$ (10) as $u (\bar{m}) \leq v (\bar{n})$ ;

$\begin{matrix} \bar{u + v} (r) \\ = {\begin{matrix} \bar{m} + \bar{n}, & r \in [0, v (\bar{n})] \\ \bar{m} + ⌊ \bar{v} (0) - r (\bar{v} (0) - \bar{v} (1)) ⌋, & r \in (v (\bar{n}), u (\bar{m})] \\ \begin{matrix} ⌊ \bar{u} (0) - r (\bar{u} (0) - \bar{u} (1)) ⌋ \\ + ⌊ \bar{v} (0) - r (\bar{v} (0) - \bar{v} (1)) ⌋ \end{matrix}, & r \in (u (\bar{m}), 1] \end{matrix} \end{matrix}$ (11) as $u (\bar{m}) > v (\bar{n})$ .

Proof. From $u = F_{I} (\underline{u} (0) |_{\underline{m}}, \underline{u} (1), \bar{u} (1), \bar{u} (0) |_{\bar{m}})$ and $v = F_{I} (\underline{v} (0) |_{\underline{n}}, \underline{v} (1), \bar{v} (1), \bar{v} (0) |_{\bar{n}})$ ), we see $\begin{matrix} u (x) & = & {\begin{matrix} \frac{x - \underline{u} (0)}{\underline{u} (1) - \underline{u} (0)} & if & x \in 〈 \underline{m}, \underline{u} (1) 〉 \\ 1 & if & x \in 〈 \underline{u} (1), \bar{u} (1) 〉 \\ \frac{\bar{u} (0) - x}{\bar{u} (0) - \bar{u} (1)} & if & x \in 〈 \bar{u} (1), \bar{m} 〉 \\ 0 & if & x \notin 〈 \underline{m}, \bar{m} 〉 \end{matrix} \\ v (x) & = & {\begin{matrix} \frac{x - \underline{v} (0)}{\underline{v} (1) - \underline{v} (0)} & if & x \in 〈 \underline{m}, \underline{v} (1) 〉 \\ 1 & if & x \in 〈 \underline{v} (1), \bar{v} (1) 〉 \\ \frac{\bar{v} (0) - x}{\bar{v} (0) - \bar{v} (1)} & if & x \in 〈 \bar{v} (1), \bar{m} 〉 \\ 0 & if & x \notin 〈 \underline{m}, \bar{m} 〉 \end{matrix} \end{matrix}$ so we have $\begin{matrix} \underline{u} (r) & = & {\begin{matrix} \underline{m} & if & r \in [0, u (\underline{m})] \\ ⌈ \underline{u} (0) + r (\underline{u} (1) - \underline{u} (0)) ⌉ & if & r \in (u (\underline{m}), 1] \end{matrix} \\ \bar{u} (r) & = & {\begin{matrix} \bar{m} & if & r \in [0, u (\bar{m})] \\ ⌊ \bar{u} (0) - r (\bar{u} (0) - \bar{u} (1)) ⌋ & if & r \in (u (\bar{m}), 1] \end{matrix} \end{matrix}$ and $\begin{matrix} \underline{v} (r) & = & {\begin{matrix} \underline{n} & if & r \in [0, v (\underline{n})] \\ ⌈ \underline{v} (0) + r (\underline{v} (1) - \underline{v} (0)) ⌉ & if & r \in (v (\underline{n}), 1] \end{matrix} \\ \bar{v} (r) & = & {\begin{matrix} \bar{n} & if & r \in [0, v (\bar{n})] \\ ⌊ \bar{v} (0) - r (\bar{v} (0) - \bar{v} (1)) ⌋ & if & r \in (v (\bar{n}), 1] \end{matrix} \end{matrix}$

Therefore, by $\underline{u + v} (r) = \underline{u} (r) + \underline{v} (r)$ and $\bar{u + v} (r) 〉 = \bar{u} (r) + \underline{v} (r)$ for any r ∈ [0, 1] (Theorem 3.4 in [15]), we have that if $u (\underline{m}) \leq v (\underline{n})$ , then

as $0 \leq r \leq u (\underline{m})$ , $\underline{u + v} (r) = \underline{m} + \underline{n}$

as $u (\underline{m}) < r \leq v (\underline{n})$ , $\underline{u + v} (r) = ⌈ \underline{u} (0) + r (\underline{u} (1) - \underline{u} (0)) ⌉ + \underline{n}$

as $v (\underline{n}) < r \leq 1$ , $\begin{matrix} \underline{u + v} (r) & = & ⌈ \underline{u} (0) + r (\underline{u} (1) - \underline{u} (0)) ⌉ \\ + ⌈ \underline{v} (0) + r (\underline{v} (1) - \underline{v} (0)) ⌉ \end{matrix}$

i.e., the Equation (8) holds. Likewise, we can see that Equations (8), (9) and (10) also hold as, respectively,

u (\underline{m}) > v (\underline{n})

u (\bar{m}) \leq v (\bar{n})

and

u (\bar{m}) > v (\bar{n})

. The proof of the theorem is completed.

5 Integral of fuzzy integer valued mapping

In this section, we give the concept of integral of fuzzy integer valued mapping based on the definition of integral of continuous fuzzy number valued mapping (see Definition 3.1 in [6]).

Theorem 3. Let u ∈ F_I and $〈 \underline{u} (0), \bar{u} (0) 〉 = {x_{0}, x_{1}, \dots, x_{m}} \cup 〈 \underline{u} (1), \bar{u} (1) 〉 \cup {y_{n}, \dots, y_{1}, y_{0}}$ with $\underline{u} (0) = x_{0} < x_{1} < \dots < x_{m} < x_{m + 1} = \underline{u} (1) \leq \bar{u} (1) = y_{n + 1} < y_{n} < \dots < y_{1} < y_{0} = \bar{u} (0)$ . If u_C : R → [0, 1] is defined as following: $u_{C} (x) = {\begin{matrix} L (x), & x \in [\underline{u} (0), \underline{u} (1)) \\ 1, & x \in [\underline{u} (1), \bar{u} (1)] \\ R (x), & x \in (\bar{u} (1), \bar{u} (0)] \\ 0, & x \notin [\underline{u} (0), \bar{u} (0)] \end{matrix}$ where L (x) is the function defined by the fold line which passes through points $(x_{0}, u (x_{0})), (x_{1}, u (x_{1})), \dots, (x_{m + 1}, u (x_{m + 1}))$ in turn, and R (x) is the function defined by the fold line which passes through points $(y_{n + 1}, u (y_{n + 1})), (y_{n}, u (y_{n})), \dots, (y_{0}, u (y_{0}))$ in turn, then u_C ∈ E (i.e., u is a continuous fuzzy number).

Proof. (1) From the definition of u_C, we can directly see u_C is a regular fuzzy set of R.

(2) Let x, y ∈ R, t ∈ [0, 1]. To prove that u is a convex fuzzy set, we only show u_C (tx + (1 - t) y) ≥ min {u_C (x) , u_C (y)} as x ≤ y (it can be similarly seen as x > y).

(i) As x, y ∈ (- ∞ , y_n+1), from the non-decrease of L (x) (it can be seen from u (x_i) ≤ u (x_i+1) , i = 0, 1, 2, ⋯ , m) and the definition of u_C, we know that u_C (x) is non-decreasing on (- ∞ , y_n+1). Therefore, from - ∞ < x ≤ tx + (1 - t) y ≤ y < y_n+1, we have u_C (tx + (1 - t) y) ≥ u_C (x) = min {u_C (x) , u_C (y)};

(ii) As x ∈ (- ∞ , y_n+1) and y ∈ [y_n+1, + ∞), (a) If tx + (1 - t) y ∈ (- ∞ , y_n+1), from the non-decrease of u_C (x) on (- ∞ , y_n+1) and - ∞ < x ≤ tx + (1 - t) y < y_n+1, we know that u_C (tx + (1 - t) y) ≥ u_C (x) ≥ min {u_C (x) , u_C (y)}; (b) If tx + (1 - t) y ∈ [y_n+1, + ∞], then from the the non-increase of u_C (x) on [y_n+1, - ∞) and y_n+1≤ tx + (1 - t) y ≤ y < + ∞, we know that u_C (tx + (1 - t) y) ≥ u_C (y) ≥ min {u_C (x) , u_C (y)}. Thus, we have proved that u_C (tx + (1 - t) y) ≥ min {u_C (x) , u_C (y)} as x ∈ (- ∞ , y_n+1) and y ∈ [y_n+1, + ∞);

(iii) As x, y ∈ [y_n+1, + ∞), from the non-increase of R (x) (it can be seen from u (y_i) ≤ u (y_i+1) , i = 0, 1, 2, ⋯ , n) and the definition of u_C, we know that u_C (x) is non-increasing on [y_n+1, + ∞). Therefore, from y_n+1≤ x ≤ tx + (1 - t) y ≤ y < + ∞, we have u_C (tx + (1 - t) y) ≥ u_C (y) = min {u_C (x) , u_C (y)}.

By the above (i)-(iii), we know that u (tx + (1 - t) y) ≥ min {u (x) , u (y)} holds, so u is a convex fuzzy set.

(3) From the continuity of u_C (x) on [x₀, y₀] (by the definition of u_C) and u_C (x) =0 as x ∉ [x₀, y₀], we can easily see u_C (x) is upper semi-continuous.

(4) It is obvious (by the definition of u_C) that the closure of suppu_C, i.e., $[u_{C}]^{0} = \bar{{x \in R : u (x) > 0}}$ is $[x_{0}, y_{0}] (= [\underline{u} (0), \bar{u} (0)])$ , which is bounded closed set, i.e., [u_C] ⁰ is compact.

By the above (1) - (4), we know that u_C is a fuzzy number. The proof of the theorem is completed.

Definition 1. For u ∈ F_I, we say the fuzzy number u_C which is defined in Theorem 3 to be the C-fuzzy number of u, and in this paper, we always use (u) _C or u_C to represent the C-fuzzy number of u.

Theorem 4. Let u ∈ E. If u_I : R → [0, 1] is defined as following: $u_{I} (x) = {\begin{matrix} u (x), & x \in 〈 ⌈ \underline{u} (0) ⌉, ⌊ \bar{u} (0) ⌋ 〉 \\ 0, & x \notin 〈 ⌈ \underline{u} (0) ⌉, ⌊ \bar{u} (0) ⌋ 〉 \end{matrix}$ then u_I ∈ F_I, and $\underline{u_{I}} (r) = ⌈ \underline{u} (r) ⌉, \bar{u_{I}} (0) = ⌊ \bar{u} (0) ⌋ for any r \in (0, 1]$ and $\begin{matrix} \underline{u_{I}} (0) & = & {\begin{matrix} ⌈ \underline{u} (0) ⌉, & u (⌈ \underline{u} (0) ⌉) \neq 0 \\ ⌈ \underline{u} (0) ⌉ + 1, & u (⌈ \underline{u} (0) ⌉) = 0 \end{matrix} \\ \bar{u_{I}} (0) & = & {\begin{matrix} ⌊ \bar{u} (0) ⌋, & u (⌊ \bar{u} (0) ⌋) \neq 0 \\ ⌊ \bar{u} (0) ⌋ - 1, & u (⌊ \bar{u} (0) ⌋) = 0 \end{matrix} \end{matrix}$

Proof. The theorem can be easily shown by the definition of u_I, so we omit it.

Definition 2. For u ∈ E, we say the fuzzy integer u_I which is defined in Theorem 4 to be the I-fuzzy integer of u, and in this paper, we always use (u) _I or u_I to represent the I-fuzzy integer of u.

Obviously, by Definitions 1 and 2, we have the following conclusion:

Conclusion 1. If u ∈ F_I and v ∈ E, then (u_C) _I = u and (v_I) _C = v.

In the following, we give the definition of integral of fuzzy integer valued mapping:

Definition 3. Let F : [a, b] → F_I be a fuzzy integer valued mapping. Define fuzzy number valued mapping F_C : [a, b] → E as F_C (t) = (F (t)) _C, ∀ t ∈ [a, b]. We say fuzzy integer valued mapping F to be integrable on [a, b] if fuzzy number valued mapping F_C is integrable on [a, b] (about the integrability of fuzzy number valued mapping, we can see Definition 3.1 in [6]), and call ${(\int_{a}^{b} F_{C} (t) dt)}_{I}$ the integral of fuzzy integer valued mapping F on [a, b], and denote it as $\int_{a}^{b} F (t) dt$ , i.e., $\int_{a}^{b} F (t) dt = {(\int_{a}^{b} F_{C} (t) dt)}_{I} .$

By Definition 3 and Conclusion 1, we can directly obtain the following conclusion:

Conclusion 2. Let F : [a, b] → F_I be a integrable fuzzy integer valued mapping, then $\int_{a}^{b} F_{C} (t) dt = {(\int_{a}^{b} F (t) dt)}_{C} .$

In the following, we give the formula for calculating the integral of fuzzy integer valued mapping in cut set:

Theorem 5. Let F : [a, b] → F_I be a integrable fuzzy integer valued mapping. Then for any r ∈ [0, 1], $\begin{matrix} \underline{(\int_{a}^{b} F (t) dt)} (r) = ⌈ \int_{a}^{b} \underline{F_{C} (t)} (r) dt ⌉ \\ \bar{(\int_{a}^{b} F (t) dt)} (r) = ⌊ \int_{a}^{b} \bar{F_{C} (t)} (r) dt ⌋ \end{matrix}$

Proof. As r ∈ (0, 1], by Definition 3, Theorem 4 and Theorem 3.2 in [6], we have $\begin{matrix} \underline{(\int_{a}^{b} F (t) dt)} (r) & = & \underline{{(\int_{a}^{b} F_{C} (t) dt)}_{I}} (r) \\ = & ⌈ \underline{\int_{a}^{b} F_{C} (t) dt} (r) ⌉ \\ = & ⌈ \int_{a}^{b} \underline{F_{C} (t)} (r) dt ⌉ \\ \bar{(\int_{a}^{b} F (t) dt)} (r) & = & \bar{{(\int_{a}^{b} F_{C} (t) dt)}_{I}} (r) \\ = & ⌊ \bar{\int_{a}^{b} F_{C} (t) dt} (r) ⌋ \\ = & ⌊ \int_{a}^{b} \bar{F_{C} (t)} (r) dt ⌋ \end{matrix}$

On the other hand, for any u ∈ F_I, by the definition of u_C and the definitions of $\underline{u} (0)$ and $\bar{u} (0)$ , we see $u_{C} (\underline{u} (0)) = u (\underline{u} (0)) \neq 0$ and $u_{C} (\bar{u} (0)) = u (\bar{u} (0)) \neq 0$ Therefore, from $\int_{a}^{b} F_{C} (t) dt \in F_{I}$ , by Conclusion 2, we know $\begin{matrix} (\int_{a}^{b} F_{C} (t) dt) (\underline{\int_{a}^{b} F_{C} (t) dt} (0)) \\ = {(\int_{a}^{b} F (t) dt)}_{C} (\underline{\int_{a}^{b} F_{C} (t) dt} (0)) \neq 0 \end{matrix}$ and $\begin{matrix} (\int_{a}^{b} F_{C} (t) dt) (\bar{\int_{a}^{b} F_{C} (t) dt} (0)) \\ = {(\int_{a}^{b} F (t) dt)}_{C} (\bar{\int_{a}^{b} F_{C} (t) dt} (0)) \neq 0 \end{matrix}$

So, by Definition 3, Theorem 4 and Theorem 3.2 in [6], we have $\begin{matrix} \underline{(\int_{a}^{b} F (t) dt)} (0) & = & \underline{{(\int_{a}^{b} F_{C} (t) dt)}_{I}} (0) \\ = & ⌈ \underline{\int_{a}^{b} F_{C} (t) dt} (0) ⌉ \\ = & ⌈ \int_{a}^{b} \underline{F_{C} (t)} (0) dt ⌉ \\ \bar{(\int_{a}^{b} F (t) dt)} (0) & = & \bar{{(\int_{a}^{b} F_{C} (t) dt)}_{I}} (0) \\ = & ⌊ \bar{\int_{a}^{b} F_{C} (t) dt} (0) ⌋ \\ = & ⌊ \int_{a}^{b} \bar{F_{C} (t)} (0) dt ⌋ . \end{matrix}$

Thus, we have proved that for any r ∈ [0, 1],

$\underline{(\int_{a}^{b} F (t) dt)} (r) = ⌈ \int_{a}^{b} \underline{F_{C} (t)} (r) dt ⌉$ and

$\bar{(\int_{a}^{b} F (t) dt)} (r) = ⌊ \int_{a}^{b} \bar{F_{C} (t)} (r) dt ⌋ .$

In order to study the operational properties of the integral of fuzzy integer valued mapping F, we first give the following two lemmas:

Lemma 1. If u, v are all trapezoid fuzzy integers and k ∈ R, then (u + v) _C and (ku) _C are all trapezoid fuzzy numbers, and we have

(u + v) _C = u_C + v_C;

(ku) _C = k · u_C.

Proof. The proof of the lemma can be easily completed by the definitions of trapezoid fuzzy integer, trapezoid fuzzy number and C-fuzzy number of fuzzy integer and the properties of corresponding operations “+” and “·”, so we omit it.

Lemma 2. If u, v are all trapezoid fuzzy numbers and k ∈ R, then (u + v) _I and (ku) _I are all trapezoid fuzzy integers, and we have

(u + v) _I = u_I + v_I;

(ku) _I = k ∘ u_I.

Proof. The proof of the lemmas can be easily completed by the definitions of trapezoid fuzzy number, trapezoid fuzzy integer and I-fuzzy integer of fuzzy number and the properties of corresponding operations “+”, “·” and “∘”, so we omit it.

Theorem 6. Let F (t) , G (t) be all integrable trapezoid fuzzy integer valued mapping on [a, b], and k, l ∈ R. Then kF (t) + lG (t) is integrable on [a, b], and

$\begin{matrix} \int_{a}^{b} (kF (t) + lG (t)) dt \\ = k \circ \int_{a}^{b} F (t) dt + l \circ \int_{a}^{b} G (t) dt \end{matrix}$

Proof. By the Definition 3, the Lemmas 1 and 2 and Theorem 3.4 in [6], we have $\begin{matrix} \int_{a}^{b} (kF (t) + lG (t)) dt \\ = {(\int_{a}^{b} (kF (t) + lG (t))_{C} dt)}_{I} \\ = {(\int_{a}^{b} ((kF)_{C} (t) + (lG)_{C} (t)) dt)}_{I} \\ = {(\int_{a}^{b} ({kF}_{C} (t) + {lG}_{C} (t)) dt)}_{I} \\ = {(k \int_{a}^{b} F_{C} (t) dt + l \int_{a}^{b} G_{C} (t) dt)}_{I} \\ = {(k \int_{a}^{b} F_{C} (t) dt)}_{I} + {(l \int_{a}^{b} G_{C} (t) dt)}_{I} \\ = k \circ {(\int_{a}^{b} F_{C} (t) dt)}_{I} + l \circ {(\int_{a}^{b} G_{C} (t) dt)}_{I} \\ = k \circ \int_{a}^{b} F (t) dt + l \circ \int_{a}^{b} G (t) dt \end{matrix}$ so the theorem holds.

Theorem 7. Let F (t) be an integrable trapezoid fuzzy integer valued mapping on [a, b], and c ∈ [a, b]. Then $\int_{a}^{b} F (t) dt = \int_{a}^{c} F (t) dt + \int_{c}^{b} G (t) dt$

Proof. By the Definition 3, the Lemmas 1 and 2 and Theorem 3.3 in [6], we have $\begin{matrix} \int_{a}^{b} F (t) dt & = & {(\int_{a}^{b} F_{C} (t) dt)}_{I} \\ = & {(\int_{a}^{c} F_{C} (t) dt + \int_{c}^{b} F_{C} (t) dt)}_{I} \\ = & {(\int_{a}^{c} F_{C} (t) dt)}_{I} + {(\int_{c}^{b} F_{C} (t) dt)}_{I} \\ = & \int_{a}^{c} F (t) dt + \int_{c}^{b} G (t) dt \end{matrix}$ so the theorem holds.

6 Application

For the convenience of writing, we use i (± j) to represent the time period from time i - 1 :60 - j to time i : j (where “** : **” is the representation of time), for example, 5 (±10) represents the time period from time 4 : 50 to time 5 : 10.

Problem: The problem to be solved is to estimate the daily traffic of a road.

1. Selection of time nodes: In order to count the traffic flow of 24 hours a day on a road, first, we randomly selected M₀ days, and record down the number of vehicles during the 2ɛ₀ (ɛ₀ is a integer with 1 ≤ ɛ₀ ≤ 4) minutes before and after each integer time of each day in the M₀ days. We use M_ij to denote the number of vehicles being counted during the time period j (± ɛ₀) (j = 1, 2, ⋯ , 24) in the ith day of the M₀ days. Thus we can obtain the following statistical data: $\begin{matrix} 1 (\pm ɛ_{0}) & 2 (\pm ɛ_{0}) & \dots & 24 (\pm ɛ_{0}) \\ M_{11} & M_{12} & \dots & M_{1 (24)} \\ M_{21} & M_{22} & \dots & M_{2 (24)} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ M_{M_{0} 1} & M_{M_{0} 2} & \dots & M_{M_{0} (24)} \end{matrix}$

By ${\bar{M}}_{j} = \frac{1}{M_{0}} \sum_{i = 1}^{M_{0}} M_{ij}$ , work out ${\bar{M}}_{j}$ (j = 1, 2, ⋯ , 24). According to the actual situation, select a proper integer threshold n₀ (Generally, traffic flow has two peaks a day, so we should usually make ${j, j + 1 : | {\bar{M}}_{j + 1} - {\bar{M}}_{j} | \geq n_{0}, j = 1, 2, 3, \dots, 23}$ have about 8 integers). If $| {\bar{M}}_{j +} - {\bar{M}}_{j} | < n_{0}$ , we consider that the vehicle speed change is not large in the time period from time j : 00 to time j + 1 :00; Conversely, If $| {\bar{M}}_{j +} - {\bar{M}}_{j} | \geq n_{0}$ , then we consider that the vehicle speed change is large in the time period from time j : 00 to time j + 1 :00. Suppose ${j_{1}, j_{2}, \dots, j_{k_{0}}} = {j, j + 1 : | {\bar{M}}_{j + 1} - {\bar{M}}_{j} | \geq n_{0}, j = 1, 2, \dots, 23}$ with j₁ < j₂ < ⋯ < j_{k
₀} (k₀ ∈ 〈2, 24〉, generally, we may usually make k₀ be about 6 by choosing integer threshold n₀). Thus, we have obtained k₀ time nodes (Note: If the interval between two adjacent two time nodes is longer, we can add some time nodes in the two time nodes).

2. Implementation of statistical method: Next, we randomly again selected N₀ days (can include the previous M₀ days, generally N₀ ≥ M₀), and respectively record down the number (denoted as N_ik) of vehicles being counted during the time period j_k (± ɛ₀) (k = 1, 2, ⋯ , k₀) in the ith day of the N₀ days. Thus we can obtain the following statistical data: $\begin{matrix} j_{1} (\pm ɛ_{0}) & j_{2} (\pm ɛ_{0}) & \dots & j_{k_{0}} (\pm ɛ_{0}) \\ N_{1 j_{1}} & M_{1 j_{2}} & \dots & M_{1 j_{k_{0}}} \\ N_{2 j_{1}} & M_{2 j_{2}} & \dots & M_{2 j_{k_{0}}} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ N_{N_{0} j_{1}} & M_{N_{0} j_{2}} & \dots & M_{N_{0} j_{k_{0}}} \end{matrix}$

Step 1: For each j_k (k = 1, 2, ⋯ , k₀), we work out the means μ_{j
_k}, the left separation degrees Lσ_{j
_k} and the right separation degrees Rσ_{j
_k} of the statistical data N_{1j_k}, N_{2j_k}, ⋯ , N_{N
₀
j
_k}: $μ_{j_{k}} = \frac{1}{N_{0}} \sum_{i = 1}^{N_{0}} N_{{ij}_{k}}$ (12) $\begin{matrix} L σ_{j_{k}} & = & \frac{1}{N_{{Lj}_{k}}} \sum_{N_{{ij}_{k}} < μ_{j_{k}}} (μ_{j_{k}} - N_{{ij}_{k}}) \\ R σ_{j_{k}} & = & \frac{1}{N_{{Rj}_{k}}} \sum_{N_{{ij}_{k}} > μ_{j_{k}}} (N_{{ij}_{k}} - μ_{j_{k}}) \end{matrix}$ (13) where N_{Lj
_k} and N_{Rj
_k} are the number of the statistical values which satisfy N_{ij
_k} ≤ μ_{j
_k} in N_{1j_k}, ⋯ , N_{N
₀
j
_k}, k = 1, 2, ⋯ , k₀ and the number of the statistical values which satisfy N_{ij
_k} ≥ μ_{j
_k} in N_{1j_k}, ⋯ , N_{N
₀
j
_k}, k = 1, 2, ⋯ , k₀, respectively.

Step 2: For each j_k (k = 1, 2, ⋯ , k₀), denote $\begin{matrix} {\underline{S}}_{0 j_{k}} & = & max {0, ⌊ μ_{j_{k}} - λ L σ_{j_{k}} ⌋} \\ = & max {0, max {n \in I : n \leq μ_{j_{k}} - λ L σ_{j_{k}}}} \\ {\underline{S}}_{1 j_{k}} & = & ⌊ μ_{j_{k}} ⌋ = max {n \in I : n \leq μ_{j_{k}}} \\ {\bar{S}}_{0 j_{k}} & = & ⌈ μ_{j_{k}} + λ R σ_{j_{k}} ⌉ \\ = & min {n \in I : n \geq μ_{j_{k}} + λ R σ_{j_{k}}} \\ {\bar{S}}_{1 j_{k}} & = & ⌈ μ_{j_{k}} ⌉ = min {n \in I : n \geq μ_{j_{k}}} \end{matrix}$ where λ is a parameter, that may be chosen in interval [2, 4] according to practical case. Obviously, the quantity of the vehicle flow during j_k (± ɛ₀) should be of Two-sided type (see Fig. 1 in [15]), so for the convenience of calculations, we can construct a trapezoid fuzzy integer u_{j
_k} as $u_{j_{k}} = F_{I} ({\underline{S}}_{0 j_{k}}, {\underline{S}}_{1 j_{k}}, {\bar{S}}_{1 j_{k}}, {\bar{S}}_{0 j_{k}})$ (14) to represent the quantity of the vehicle flow during j_k (± ɛ₀). Thus, we can use trapezoid fuzzy integer $\begin{matrix} v_{j_{k}} = \frac{60}{2 ɛ_{0}} \circ u_{j_{k}} & = & F_{I} (⌊ \frac{30}{ɛ_{0}} {\underline{S}}_{0 j_{k}} ⌉, ⌊ \frac{30}{ɛ_{0}} {\underline{S}}_{1 j_{k}} ⌉, \\ ⌊ \frac{30}{ɛ_{0}} {\bar{S}}_{1 j_{k}} ⌉, ⌊ \frac{30}{ɛ_{0}} {\bar{S}}_{0 j_{k}} ⌉) \end{matrix}$ (Unit: vehicle/hour) to represent the speed of vehicle flow at time j_k.

Step 3: We construct a fold line type fuzzy integer valued mapping v = v (t) to approximate the speed of vehicle flow at time t in all-day as following:

$\begin{matrix} v (t) \\ = {\begin{matrix} \frac{j_{2} - t}{j_{2} - j_{1}} \circ v_{j_{1}} + \frac{t - j_{1}}{j_{2} - j_{1}} \circ v_{j_{2}}, & t \in [j_{1}, j_{2}) \\ \frac{j_{3} - t}{j_{3} - j_{2}} \circ v_{j_{2}} + \frac{t - j_{2}}{j_{3} - j_{2}} \circ v_{j_{3}}, & t \in [j_{2}, j_{3}) \\ ⋮ \\ \frac{j_{1} + 24 - t}{j_{1} + 24 - j_{k_{0}}} \circ v_{j_{k_{0}}} + \frac{t - j_{k_{0}}}{j_{1} + 24 - j_{k_{0}}} \circ v_{j_{1}}, & t \in [j_{k_{0}}, j_{1} + 24] \end{matrix} \end{matrix}$ (15)

Step 4: We work out the integral $\int_{j_{1}}^{j_{1} + 24} v (t) dt$ , then the integral value can be used to approximate the quantity of daily traffic of the road.

Example. Suppose we have sought out the time nodes are 6, 7, 9, 10, 16, 17, 20, 21 according to the method of Selection of time nodes introduced above. We notice that the interval between time 21 and time 6 (of the next day) is longer, so we add a time nodes 1. Thus we have j₁ = 1, j₂ = 6, j₃ = 7, j₄ = 9, j₅ = 10, j₆ = 16, j₇ = 17, j₈ = 20, j₉ = 21. And suppose we have obtained the following statistical data (where taking ɛ₀ = 3):

$\begin{matrix} 0 : 57 - 1 : 03 & 5 : 57 - 6 : 03 & 6 : 57 - 7 : 03 & 8 : 57 - 9 : 03 & 9 : 57 - 10 : 03 \\ 2 & 22 & 51 & 66 & 36 \\ 4 & 32 & 45 & 35 & 40 \\ 3 & 18 & 69 & 61 & 30 \\ 8 & 32 & 54 & 35 & 54 \\ 3 & 24 & 39 & 50 & 28 \\ 5 & 34 & 45 & 53 & 39 \\ 3 & 22 & 39 & 40 & 68 \\ 9 & 14 & 69 & 45 & 42 \\ 5 & 5 & 39 & 59 & 27 \\ 7 & 19 & 24 & 77 & 39 \\ 8 & 28 & 45 & 20 & 30 \\ 5 & 18 & 66 & 35 & 27 \\ 6 & 14 & 57 & 68 & 57 \\ 5 & 16 & 51 & 41 & 33 \\ 8 & 10 & 75 & 50 & 30 \\ 2 & 15 & 39 & 65 & 24 \\ 6 & 18 & 50 & 83 & 39 \\ 4 & 20 & 36 & 24 & 17 \\ 5 & 19 & 30 & 35 & 20 \\ 3 & 26 & 42 & 50 & 39 \end{matrix}$

$\begin{matrix} 15 : 57 - 16 : 03 & 16 : 57 - 17 : 03 & 19 : 57 - 20 : 03 & 20 : 57 - 21 : 03 \\ 15 & 50 & 69 & 26 \\ 27 & 68 & 63 & 28 \\ 12 & 74 & 54 & 18 \\ 48 & 71 & 57 & 30 \\ 57 & 20 & 45 & 14 \\ 18 & 65 & 42 & 20 \\ 45 & 38 & 48 & 28 \\ 39 & 86 & 63 & 20 \\ 51 & 59 & 51 & 30 \\ 33 & 60 & 48 & 22 \\ 45 & 71 & 39 & 18 \\ 51 & 62 & 36 & 36 \\ 27 & 59 & 54 & 32 \\ 12 & 47 & 69 & 27 \\ 33 & 29 & 60 & 32 \\ 39 & 44 & 63 & 36 \\ 21 & 50 & 48 & 31 \\ 33 & 62 & 48 & 26 \\ 27 & 65 & 60 & 22 \\ 27 & 68 & 69 & 24 \end{matrix}$

Step 1: By Formulas 12 and 13, we can work out μ_{j
_k}, Lσ_{j
_k}, Rσ_j-k, k = 1, 2, ⋯ , 9 as follows: $\begin{matrix} \begin{matrix} μ_{1} & μ_{6} & μ_{7} & μ_{9} & μ_{10} & μ_{16} & μ_{17} & μ_{20} & μ_{21} \\ 5.05 & 20.3 & 48.25 & 49.6 & 35.95 & 33 & 55.9 & 54.3 & 26 \end{matrix} \\ \begin{matrix} L σ_{1} & L σ_{6} & L σ_{7} & L σ_{9} & L σ_{10} & L σ_{16} & L σ_{17} & L σ_{20} & L σ_{21} \\ 2.05 & 5.21 & 9.80 & 15.16 & 9.35 & 9.25 & 16.03 & 7.66 & 5 \end{matrix} \\ \begin{matrix} R σ_{1} & R σ_{6} & R σ_{7} & R σ_{9} & R σ_{10} & R σ_{16} & R σ_{17} & R σ_{20} & R σ_{21} \\ 1.37 & 6.37 & 11.97 & 12.4 & 9.35 & 10.09 & 10.68 & 9.37 & 4.17 \end{matrix} \end{matrix}$

Step 2: Taking λ = 2, we have $\begin{matrix} {\underline{S}}_{01} & = & max {0, max {n \in I : n \leq μ_{1} - 2 L σ_{1}}} \\ = & 0 \\ {\underline{S}}_{11} & = & max {n \in I : n \leq μ_{1}} = 5 \\ {\bar{S}}_{01} & = & min {n \in I : n \geq μ_{1} + 2 R σ_{1}} = 8 \\ {\bar{S}}_{11} & = & min {n \in I : n \geq μ_{1}} = 6 \end{matrix}$

so we can construct trapezoid fuzzy integer u₁ as $u_{1} = F_{I} (0, 5, 6, 8)$

Likewise, we can construct trapezoid fuzzy integers u_{j
_k}, j_k = 6, 7, 9, 10, 16, 17, 20, 21 as $\begin{matrix} u_{6} & = & F_{I} (9, 20, 21, 34), u_{7} = F_{I} (28, 48, 49, 73) \\ u_{9} & = & F_{I} (19, 49, 50, 75), u_{10} = F_{I} (17, 35, 36, 55) \\ u_{16} & = & F_{I} (14, 33, 33, 54), u_{17} = F_{I} (23, 55, 56, 78) \\ u_{20} & = & F_{I} (38, 54, 55, 74), u_{21} = F_{I} (16, 26, 26, 35) \end{matrix}$

By $\begin{matrix} v_{j_{k}} & = & \frac{60}{6} \circ u_{j_{k}} \\ = & F_{I} (⌊ 10 {\underline{S}}_{0 j_{k}} ⌉, ⌊ 10 {\underline{S}}_{1 j_{k}} ⌉, ⌊ 10 {\bar{S}}_{1 j_{k}} ⌉, ⌊ 10 {\bar{S}}_{0 j_{k}} ⌉) \end{matrix}$ j_k = 1, 6, 7, 9, 10, 16, 17, 20, 21, we can obtain $\begin{matrix} v_{1} = F_{I} (0, 50, 60, 80), \\ v_{6} = F_{I} (90, 200, 210, 340) \\ v_{7} = F_{I} (280, 480, 490, 730), \\ v_{9} = F_{I} (190, 490, 500, 750) \\ v_{10} = F_{I} (170, 350, 360, 550), \\ v_{16} = F_{I} (140, 330, 330, 540) \\ v_{17} = F_{I} (230, 550, 560, 780), \\ v_{20} = F_{I} (380, 540, 550, 740) \\ v_{21} = F_{I} (160, 260, 260, 350) \end{matrix}$ and use v_{j
_k} to represent the speed of vehicle flow at time j_k (Unit: vehicle/hour), j_k = 1, 6, 7, 9, 10, 16, 17, 20, 21.

Step 3: By Formula (15), we can construct the following fold line type fuzzy integer valued mapping v = v (t) to approximate the speed of vehicle flow at time t in all-day as following:

$\begin{matrix} v (t) \\ = {\begin{matrix} \frac{6 - t}{5} \circ v_{1} + \frac{t - 1}{5} \circ v_{6}, & t \in [1, 6) \\ (7 - t) \circ v_{6} + (t - 6) \circ v_{7}, & t \in [6, 7) \\ ⋮ \\ \frac{25 - t}{4} \circ v_{21} + \frac{t - 21}{4} \circ v_{1}, & t \in [21, 25] \end{matrix} \end{matrix}$ (16)

Step 4: We work out the integral $\begin{matrix} \int_{1}^{25} v (t) dt \\ = {(\int_{1}^{25} v_{C} (t) dt)}_{I} \\ = (\int_{1}^{6} v_{C} (t) dt + \int_{6}^{7} v_{C} (t) dt + \int_{7}^{9} v_{C} (t) dt \\ + \int_{9}^{10} v_{C} (t) dt + \int_{10}^{16} v_{C} (t) dt + \int_{16}^{17} v_{C} (t) dt \\ {+ \int_{17}^{20} v_{C} (t) dt + \int_{20}^{21} v_{C} (t) dt + \int_{21}^{25} v_{C} (t) dt)}_{I} \\ = [(225, 625, 675, 875) + (55, 340, 350, 355) \\ + (470, 970, 990, 1480) \\ + (180, 420, 430, 650) \\ + (1710, 2040, 2070, 5370) \\ + (185, 440, 445, 660) \\ + (915, 1635, 1665, 2280) \\ + (270, 400, 405, 545) \\ + (320, 620, 640, 860)]_{I} \\ = (4330, 7490, 7670, 13075)_{I} \\ = F_{I} (4330, 7490, 7670, 13075) \end{matrix}$ then the integral value F_I (4330, 7490, 7670, 13075) can be used to approximate the quantity of daily traffic of the road.

7 Conclusion

In this paper, on the basis of the methods of constructing fuzzy integer, we established a method (Section 3) for constructing fold line type fuzzy integer valued mapping to represent the change rule of a uncertain integer quantity which changes with the change of some quantity by using statistical method. Then we provided the calculation formulas (in cut set, Theorems 1 and 2) of linear operations of trapezoid-type fuzzy integers for the convenience of calculations in applications. And then, by giving the definition of continuizing of fuzzy integer (Theorem 3 and Definition 1) and integerizing of continuous fuzzy number (Theorem 4 and Definition 2), we gave a definition (Definition 3) of integral of fuzzy integer valued mapping based on the definition of integral of continuous fuzzy number valued mapping, and obtained its calculating formula in cut set (Theorem 5) and property theorems (Theorems 6 and 7). Finally, taking the case of estimating the quantity of daily vehicle flow of a road, we proposed a method (Section 6) for obtaining some imprecise integer statistical quantities based on the theory set up by us, and gave a practical example (Example 1) to show the specific algorithm of this method.

Footnotes

Acknowledgments

This work is partially supported by the Nature Science Foundation of China (No. 61433001).

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