Models for software quality evaluation with fuzzy number intuitionistic fuzzy information

Abstract

Software is omnipresent in our daily lives. Delivering quality software on schedule and in budget has become the challenge for most companies. As software size and complexity are increasing at a geometric rate, assuring software quality is becoming more difficult. Thus, how to control and manage software quality has attracted more and more attentions. In this paper, we investigate the multiple attribute decision making problems with fuzzy number intuitionistic fuzzy information. We first introduce some operations on the fuzzy number intuitionistic fuzzy sets. Then, we further develop the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator. Then, we apply the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator to deal with multiple attribute decision making under the fuzzy number intuitionistic fuzzy environments. Finally, an illustrative example for evaluating the software quality is given to verify the developed approach.

Keywords

Multiple attribute decision making (MADM)fuzzy number intuitionistic fuzzy sets Hamacher aggregation operators fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator software quality evaluation

1 Introduction

With the increasing popularity of software products, the market requires high software product quality continuously. How to evaluate the quality of software has become the most concerned problem to users and managers of software organizations, because the evaluation results of software quality can not only guide the users to purchase and use the software, but also guide software developers to develop high-quality software products [1 –3]. Previous studies have shown that, due to the characteristics of the software itself and the limitation of public cognitive level, software quality evaluations are always vague and uncertain. Just because of the fuzzy characteristics of software quality, domestic and overseas scholars began to use fuzzy comprehensive evaluation method to solve some core issues of software quality evaluation [4 –8]. In the fuzzy comprehensive evaluation method, membership function construction is a quite difficult and complex process, and the construct of the weight is not reasonable. In the construction of membership function, in order to meet the continuous and gradient characteristics of quality, semitrapezoid distribution and trapezoidal distribution were currently used [6 –11].

In this paper, we investigate the multiple attribute decision making problems with fuzzy number intuitionistic fuzzy information. We first introduce some operations on the fuzzy number intuitionistic fuzzy sets. Then, we further develop the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator. Then, we apply the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator to deal with multiple attribute decision making under the fuzzy number intuitionistic fuzzy environments. To do so, the remainder of this paper is set out as follows. In the next section, we introduce some basic concepts related to fuzzy number intuitionistic fuzzy sets. In Section 3 we have developed the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator. In Section 4, we have applied the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator to develop the model for multiple attribute decision making problems with fuzzy number intuitionistic fuzzy information. In Section 5, a practical example for evaluating the software quality is given to verify the developed approach and to demonstrate its practicality and effectiveness. In Section 6, we conclude the paper and give some remarks.

2 Preliminaries

Atanassov [12, 13] introduced the concept of intuitionistic fuzzy set(IFS), which is a generalization of the concept of fuzzy set [14]. Each element in the IFs is expressed by an ordered pair, and each ordered pair is characterized by a membership degree and a non-membership degree. Recently, many works have investigated the intuitionistic fuzzy sets [15 –37]. Liu and Yuan [38] introduced the concept of fuzzy number intuitionistic fuzzy set(FNIFS) which fundamental characteristic of the FNIFS is that the values of its membership function and non-membership function are triangular fuzzy numbers rather than exact numbers. Wang [39] developed the fuzzy number intuitionistic fuzzy weighted averaging (FNIFWA) operator, fuzzy number intuitionistic fuzzy ordered weighted averaging (FNIFOWA) operator and fuzzy number intuitionistic fuzzy hybrid aggregation (FNIFHA) operator. Wang [40] proposed some aggregation operators, including fuzzy number intuitionistic fuzzy weighted geometric (FNIFWG) operator, fuzzy number intuitionistic fuzzy ordered weighted geometric (FNIFOWG) operator and fuzzy number intuitionistic fuzzy hybrid geometric (FNIFHG) operator. Lin et al. [41] developed the fuzzy number intuitionistic fuzzy prioritized operators and their application to multiple attribute decision making.

Definition 1. [38] Given a fixed set X ={ x₁, x₂, …, x_n }, An FNIFS $\tilde{A}$ over X is an object having the form: $\tilde{A} = {〈 x_{i}, {\tilde{t}}_{A} (x_{i}), {\tilde{f}}_{A} (x_{i}) 〉 | x_{i} \in X}$ (1) where ${\tilde{t}}_{A} (x_{i}) \subset [0, 1]$ and ${\tilde{f}}_{A} (x_{i}) \subset [0, 1]$ are triangular fuzzy numbers, and $\begin{matrix} {\tilde{t}}_{A} (x_{i}) = (a (x_{i}), b (x_{i}), c (x_{i})), X \to [0, 1], \\ {\tilde{f}}_{A} (x) = (l (x_{i}), m (x_{i}), p (x_{i})), X \to [0, 1], \\ 0 \leq c (x_{i}) + p (x_{i}) \leq 1, \forall x \in X . \end{matrix}$

For convenience, let ${\tilde{t}}_{A} (x_{i}) = (a (x_{i}), b (x_{i}), c (x_{i})),$ ${\tilde{f}}_{A} (x) = (l (x_{i}), m (x_{i}), p (x_{i}))$ , so $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ and we call $\tilde{a} (x_{i})$ an fuzzy number intuitionistic fuzzy value (FNIFV).

Definition 2. [39] Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})),$ (l (x_i) , m (x_i) , p (x_i))〉 be a collection of FNIFEs, a score function S of a FNIFV $\tilde{a} (x_{i})$ can be represented as follows:

$\begin{matrix} S (\tilde{a} (x_{i})) & = & \frac{a (x_{i}) + 2 b (x_{i}) + c (x_{i})}{4} \\ - \frac{l (x_{i}) + 2 m (x_{i}) + p (x_{i})}{4} \end{matrix}$ (2)

Definition 3. [39] Let $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})),$ (l (x_i) , m (x_i) , p (x_i))〉 be a collection of FNIFEs, an accuracy function H of a FNIFV $\tilde{a} (x_{i})$ can be represented as follows [33, 34]: $H (\tilde{a} (x_{i})) = \frac{(a (x_{i}) + 2 b (x_{i}) + c (x_{i})) + (l (x_{i}) + 2 m (x_{i}) + p (x_{i}))}{4}, H (\tilde{a} (x_{i})) \in [0, 1] .$ (3) to evaluate the degree of accuracy of theFNIFV $\tilde{a} (x_{i}) = 〈 (a (x_{i}), b (x_{i}), c (x_{i})), (l (x_{i}), m (x_{i}), p (x_{i})) 〉$ , where $H (\tilde{a} (x_{i})) \in [0, 1]$ . The larger the value of $H (\tilde{a} (x_{i}))$ , the more the degree of accuracy of the FNIFV $\tilde{a} (x_{i})$ is.

3 Fuzzy number intuitionistic fuzzy Hamacher power weighted average operator

Based on the Hamacher operation [42, 43], Zhou & Chang [44] proposed some new Hamacher aggregation operators with fuzzy number intuitionistic fuzzy information, such as the fuzzy number intuitionistic fuzzy Hamacher weighted average (FNIFHWA)operator, fuzzy number intuitionistic fuzzy Hamacher ordered weighted average (FNIFHOWA) operator and fuzzy number intuitionistic fuzzy Hamacher hybrid average (FNIFHHA) operator.

Definition 4. Let $\tilde{a} (x_{j}) = 〈 (a (x_{j}), b (x_{j}), c (x_{j})),$ (l (x_j) , m (x_j) , p (x_j))〉 be a collection of FNIFEs, then we define the fuzzy number intuitionistic fuzzy Hamacher weighted average (FNIFHWA) operator as follows: $\begin{matrix} FNIFHWA (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) = \oplus_{j = 1}^{n} (ω_{j} \tilde{a} (x_{j})) \\ = 〈 (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a (x_{j}))}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - a (x_{j}))}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a (x_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - a (x_{j}))}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b (x_{j}))}^{ω_{j}} - \prod_{j = 1}^{n} (1 - γ_{j}) b {(x_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b (x_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - b (x_{j}))}^{ω_{j}}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c (x_{j}))}^{ω_{j}} - \prod_{j = 1}^{n} {(1 - c (x_{j}))}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c (x_{j}))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} {(1 - c (x_{j}))}^{ω_{j}}}), \\ (\frac{γ \prod_{j = 1}^{n} l {(x_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - l (x_{j})))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} l {(x_{j})}^{ω_{j}}}, \\ \frac{γ \prod_{j = 1}^{n} m {(x_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - m (x_{j})))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} m {(x_{j})}^{ω_{j}}}, \end{matrix} \frac{γ \prod_{j = 1}^{n} p {(x_{j})}^{ω_{j}}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - p (x_{j})))}^{ω_{j}} + (γ - 1) \prod_{j = 1}^{n} p {(x_{j})}^{ω_{j}}}) 〉$ (4)

where ω = (ω₁, ω₂, …, ω_n) ^T be the weight vector of $\tilde{a} (x_{j}) (j = 1, 2, \dots, n)$ , and ω_j > 0, $\sum_{j = 1}^{n} ω_{j} = 1$ .

Yager [45] developed a nonlinear weighted average aggregation operator called power average (PA) operator, which can be defined as follows: $PA (a_{1}, a_{2}, \dots, a_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (a_{i})) a_{i}}{\sum_{i = 1}^{n} (1 + T (a_{i}))}$ (5) where $T (a_{i}) = \sum_{j = 1 j \neq i}^{n} Sup (a_{i}, a_{j})$ , and Sup (a, b) is the support for a from b, which satisfies the following three properties:

Sup (a, b) ∈ [0, 1];

Sup (a, b) = Sup (b, a);

Sup (a, b) ≥ Sup (x, y) , if |a − b| < |x − y| .

Obvoiusly, the support (Sup) measure is essentially a similarity index. The more similar, the closer two values, and the more they support each other.

The power average [45] operators, however, have usually been used in situations where the input arguments are the exact values. In this Section, we shall investigate the PA operator under fuzzy number intuitionistic fuzzy environments. Based on equation (5), we give the definition of the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator as follows.

Definition 5. Let $\tilde{a} (x_{j}) = 〈 (a (x_{j}), b (x_{j}), c (x_{j})),$ (l (x_j) , m (x_j) , p (x_j))〉 be a collection of FNIFEs, ω = (ω₁, ω₂, …, ω_n) ^T be the weighting vector of FNIFEs ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ and ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ , then we define the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) as follows:

$\begin{matrix} FNIFHPWA (\tilde{a} (x_{1}), \tilde{a} (x_{2}), \dots, \tilde{a} (x_{n})) \\ = \frac{ω_{1} (1 + T ({\tilde{a}}_{1})) {\tilde{a}}_{1} \oplus ω_{2} (1 + T ({\tilde{a}}_{2})) {\tilde{a}}_{2} \oplus \dots \oplus ω_{n} (1 + T ({\tilde{a}}_{n})) {\tilde{a}}_{n}}{\sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))} = \oplus_{j = 1}^{n} (\frac{ω_{j} (1 + T ({\tilde{a}}_{j})) {\tilde{a}}_{j}}{\sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{j}))}) \\ = 〈 (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} - \prod_{j = 1}^{n} {(1 - a (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - a (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} - \prod_{j = 1}^{n} (1 - γ_{j}) b {(x_{j})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - b (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} - \prod_{j = 1}^{n} {(1 - c (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - c (x_{j}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}), \\ (\frac{γ \prod_{j = 1}^{n} l {(x_{j})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - l (x_{j})))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} l {(x_{j})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}, \\ \frac{γ \prod_{j = 1}^{n} m {(x_{j})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - m (x_{j})))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} m {(x_{j})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}, \\ \frac{γ \prod_{j = 1}^{n} p {(x_{j})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - p (x_{j})))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} p {(x_{j})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}) 〉 \end{matrix}$ (6)

where $T ({\tilde{α}}_{j}) = \sum_{i = 1 i \neq j}^{n} ω_{i} Sup ({\tilde{α}}_{j}, {\tilde{α}}_{i})$ (7) and $Sup ({\tilde{α}}_{j}, {\tilde{α}}_{i})$ is the support for ${\tilde{α}}_{j}$ from ${\tilde{α}}_{i}$ , with the conditions:

$Sup ({\tilde{α}}_{i}, {\tilde{α}}_{j}) \in [0, 1]$ ;

$Sup ({\tilde{α}}_{i}, {\tilde{α}}_{j}) = Sup ({\tilde{α}}_{i}, {\tilde{α}}_{j})$ ;

$Sup ({\tilde{α}}_{i}, {\tilde{α}}_{j}) \geq Sup ({\tilde{α}}_{s}, {\tilde{α}}_{t})$ , if d (α_i, α_j) ≥ d (α_s, α_t), where d is a distance measure.

It can be easily proved that the FNIFHPWA operator has the following properties.

Theorem 1.(Idempotency) Let ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ be a collection of FNIF Es,ω = (ω₁, ω₂, …, ω_n) ^Tbe the weighting vector of picture fuzzy numbers ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ andω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ . If all ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ are equal, i.e. ${\tilde{α}}_{j} = \tilde{α}$ for all j, then $FNIFHPWA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) = \tilde{α}$ (8)

Theorem 2.(Boundedness) Let ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ be a collection of FNIF Es,ω = (ω₁, ω₂, …, ω_n) ^Tbe the weighting vector of picture fuzzy numbers ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ , and let ${\tilde{α}}^{-} = min_{j} {\tilde{α}}_{j}$ , ${\tilde{α}}^{+} = max_{j} {\tilde{α}}_{j}$

Then ${\tilde{α}}^{-} \leq FNIFHPWA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \leq {\tilde{α}}^{+}$ (9)

Theorem 3.(Monotonicity) Let ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ and ${\tilde{α}}_{j}^{'} (j = 1, 2, \dots, n)$ be two collection of FNIF Es, ω = (ω₁, ω₂, …, ω_n) ^Tbe the weighting vector of picture fuzzy numbers ${\tilde{α}}_{j} (j = 1, 2, \dots, n)$ andω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ , if ${\tilde{α}}_{j} \leq {\tilde{α}}_{j}^{'}$ , for allj, then

$\begin{matrix} FNIFHPWA ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{n}) \\ \leq FNIFHPWA ({\tilde{α}}_{1}^{'}, {\tilde{α}}_{2}^{'}, \dots, {\tilde{α}}_{n}^{'}) \end{matrix}$ (10)

4 Model for multiple attribute decision making with fuzzy number intuitionistic fuzzy information

Let A ={ A₁, A₂, …, A_m } be a discrete set of alternatives, and G ={ G₁, G₂, …, G_n } be the set of attributes, ω = (ω₁, ω₂, …, ω_n) is the weighting vector of the attribute G_j (j = 1, 2, …, n), where ω_j ∈ [0, 1], $\sum_{j = 1}^{n} ω_{j} = 1$ . Suppose that $\tilde{R} = {({\tilde{r}}_{ij})}_{m \times n} = {〈 (a_{ij}, b_{ij}, c_{ij}), (l_{ij}, m_{ij}, p_{ij}) 〉}_{m \times n}$ is the fuzzy number intuitionistic fuzzy decision matrix, i = 1, 2, …, m, j = 1, 2, …, n.

In the following, we apply the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator to MADM problems for with fuzzy number intuitionistic fuzzy information.

Step 1. Utilize the decision information given in matrix $\tilde{R}$ , and the FNIFHPWA operator.

$\begin{matrix} {\tilde{r}}_{i} = {FNIFHPWA}_{ω} ({\tilde{r}}_{i 1}, {\tilde{r}}_{i 2}, \dots, {\tilde{r}}_{in}) = \oplus_{j = 1}^{n} (\frac{ω_{j} (1 + T ({\tilde{a}}_{ij})) {\tilde{a}}_{ij}}{\sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{a}}_{ij}))}) \\ = 〈 (\frac{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} - \prod_{j = 1}^{n} {(1 - a_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) a_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - a_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}, \\ \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} - \prod_{j = 1}^{n} {(1 - b_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) b_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - b_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}} \end{matrix} \begin{matrix} \frac{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} - \prod_{j = 1}^{n} {(1 - c_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) c_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(1 - c_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}), \\ (\frac{γ \prod_{j = 1}^{n} {(l_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - l_{ij}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(l_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}, \\ \frac{γ \prod_{j = 1}^{n} {(m_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - m_{ij}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(m_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}, \\ \frac{γ \prod_{j = 1}^{n} {(p_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}{\prod_{j = 1}^{n} {(1 + (γ - 1) (1 - p_{ij}))}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))} + (γ - 1) \prod_{j = 1}^{n} {(p_{ij})}^{ω_{j} (1 + T ({\tilde{h}}_{j})) / \sum_{j = 1}^{n} ω_{j} (1 + T ({\tilde{h}}_{j}))}}) 〉 \end{matrix}$ to derive the overall preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ of the alternative A_i.

Step 2. Calculate the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ of the overall fuzzy number intuitionistic fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, m)$ to rank all the alternatives A_i (i = 1, 2, …, m) and then to select the best one(s).

Step 3. Rank all the alternatives A_i (i = 1, 2, …, m) and select the best one(s) in accordance with $S ({\tilde{r}}_{i})$ and $H ({\tilde{r}}_{i}) (i = 1, 2, \dots, m)$ .

Step 4. End.

5 Numerical example

With the increasing trend of informatization, software products have reached wider application range, so that the social needs and dependence to software are stronger. Whilst, it was shown that the software system is approaching scale production and software researching techniques appear a complicated tendency, therefore, more strict requirements have been drilled out on quality measurement and quality control in regarding to the software developing processes implemented by the domestic software companies. However, due to the factors of weak consciousness of quality management, lag back of quality management system and insufficient training, it is quite common for domestic software products suffering from low quality and unsatisfied customer needs. Therefore, it is the absolute path way for those software companies to make further improvement on ability of software development and quality control, in order to meet the booming trends on information system demands. Hence, how to control and improve software process and product quality has theoretical and practical significance. In this section, we present an empirical case study of evaluating the software quality. The project’s aim is to evaluate the best software quality from the different software systems, which provide alternatives of software systems to university. The software quality of five possible software systems A_i (i = 1, 2, 3, 4, 5) is evaluated. A software selection problem can be calculated as a multiple attribute group decision making problem in which alternatives are the software packages to be selected and criteria are those attributes under consideration. A computer center in a university desires to select a new information system in order to improve work productivity. After preliminary screening, five software systems A_i (i = 1, 2, …, 5) have remained in the candidate list. Three decision makers (experts) form a committee to act as decision makers. The computer center in the university must take a decision according to the following four attributes: ➀G₁ is the costs of hardware/software investment; ➁G₂ is the contribution to organization performance; ➂G₃ is the effort to transform from current system; ➃G₄ is the outsourcing software developer reliability. The five possible software systems A_i (i = 1, 2, …, 5) are to be evaluated using the fuzzy number intuitionistic fuzzy values by the decision maker under the above four attributes, and construct, respectively, the decision matrices as listed in the following matrices $\tilde{R} = {({\tilde{r}}_{i j})}_{5 \times 4}$ as follows:

$\begin{matrix} \tilde{R} = [\begin{matrix} 〈 (0.2, 0.3, 0.4), (0.3, 0.4, 0.4) 〉 & 〈 (0.5, 0.6, 0.7), (0.1, 0.1, 0.1) 〉 \\ 〈 (0.3, 0.4, 0.5), (0.1, 0.2, 0.3) 〉 & 〈 (0.4, 0.5, 0.5), (0.1, 0.1, 0.2) 〉 \\ 〈 (0.1, 0.2, 0.3), (0.3, 0.4, 0.5) 〉 & 〈 (0.3, 0.4, 0.5), (0.2, 0.2, 0.3) 〉 \\ 〈 (0.4, 0.5, 0.6), (0.1, 0.1, 0.1) 〉 & 〈 (0.7, 0.7, 0.7), (0.1, 0.1, 0.1) 〉 \\ 〈 (0.6, 0.6, 0.7), (0.1, 0.1, 0.1) 〉 & 〈 (0.4, 0.4, 0.4), (0.1, 0.2, 0.3) 〉 \end{matrix} \\ \begin{matrix} 〈 (0.5, 0.5, 0.6), (0.1, 0.1, 0.2) 〉 & 〈 (0.4, 0.5, 0.6), (0.1, 0.1, 0.1) 〉 \\ 〈 (0.3, 0.4, 0.5), (0.1, 0.2, 0.3) 〉 & 〈 (0.1, 0.2, 0.3), (0.3, 0.4, 0.5) 〉 \\ 〈 (0.6, 0.7, 0.8), (0.1, 0.1, 0.1) 〉 & 〈 (0.1, 0.1, 0.2), (0.4, 0.5, 0.6) 〉 \\ 〈 (0.4, 0.5, 0.5), (0.1, 0.2, 0.3) 〉 & 〈 (0.3, 0.4, 0.5), (0.2, 0.3, 0.3) 〉 \\ 〈 (0.6, 0.6, 0.6), (0.1, 0.1, 0.1) 〉 & 〈 (0.2, 0.3, 0.3), (0.3, 0.4, 0.5) 〉 \end{matrix}] \end{matrix}$

In the following, we apply the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator to MADM problems for evaluating the software quality with fuzzy number intuitionistic fuzzy information.

Step 1. We utilize the decision information given in matrix $\tilde{R}$ , and the FNIFHPWA operator to obtain the overall preference values ${\tilde{r}}_{i}$ of the software systems A_i (i = 1, 2, …, 5). $\begin{matrix} {\tilde{r}}_{1} = 〈 (0.388, 0.471, 0.575), (0.257, 0.316, 0.352) 〉 \\ {\tilde{r}}_{2} = 〈 (0.232, 0.314, 0.417), (0.331, 0.433, 0.540) 〉 \\ {\tilde{r}}_{3} = 〈 (0.207, 0.274, 0.364), (0.418, 0.499, 0.602) 〉 \\ {\tilde{r}}_{4} = 〈 (0.386, 0.485, 0.549), (0.276, 0.366, 0.398) 〉 \\ {\tilde{r}}_{5} = 〈 (0.376, 0.447, 0.460), (0.302, 0.367, 0.453) 〉 \end{matrix}$

Step 2. Calculate the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, 5)$ of the overall fuzzy number intuitionistic fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5)$ $\begin{matrix} S ({\tilde{r}}_{1}) = 0.166, S ({\tilde{r}}_{2}) = - 0.115 \\ S ({\tilde{r}}_{3}) = - 0.225, S ({\tilde{r}}_{4}) = 0.125 \\ S ({\tilde{r}}_{5}) = 0.060 \end{matrix}$

Step 3. Rank all the software systems A_i (i = 1, 2, 3, 4, 5) in accordance with the scores $S ({\tilde{r}}_{i}) (i = 1, 2, \dots, 5)$ of the overall fuzzy number intuitionistic fuzzy preference values ${\tilde{r}}_{i} (i = 1, 2, \dots, 5)$ : A₁ ≻ A₄ ≻ A₅ ≻ A₂ ≻ A₃, and thus the most desirable software systems is A₁.

6 Conclusion

With the widespread use of computer and rapid development of network technology, software has penetrated into all fields of the national economy as well as national defense construction, and been a very important part of human life. Followed by the increasing the size and complexity of software, it is very difficult to control the risk of software and is very frequent that the disastrous events have happened due to software defects or failures. Development of High Confidence Software has become a strategic high ground of the current international software technology development, which has caused a widespread concern in the human society. Trustworthy software evaluation based on multi-quality-attributes becomes the core foundation to achieve software trustworthiness and carry out the management of trustworthy software, which is also an urgently resolved problem in the process of trustworthy software development and management. Currently, the major research on quality attributes of software is based on the developer’s perspective, ignoring the objective practice and subjective experience of software users in the process of using software. So the research based on user requirements in the field of trustworthy software has become very important for theory and practice. In this paper, we investigate the multiple attribute decision making problems with fuzzy number intuitionistic fuzzy information. We first introduce some operations on the fuzzy number intuitionistic fuzzy sets. Then, we further develop the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator. Then, we apply the fuzzy number intuitionistic fuzzy Hamacher power weighted average (FNIFHPWA) operator to deal with multiple attribute decision making under the fuzzy number intuitionistic fuzzy environments. Finally, an illustrative example for evaluating the software quality is given to verify the developed approach. In the future, we shall extend the proposed models to other domains [46 –53].

References

Boehm

B.W.

, Brown

J.R.

, and Lipow

, Quantitative evaluation of software quality, in Proceedings of the of the 2nd International Conference on Software Engineering1976, 592–605.

Sarkar

, Rama

G.M.

, and Kak

A.C.

, API-based and information-theoretic metrics for measuring the quality of software modularization, IEEE Transactions on Software Engineering33(1) (2007), 14–32.

Yue

, Su

, Lu

, and Zhang

, Comprehensive evaluation of software quality based on fuzzy soft sets, Systems Engineering and Electronics35(7) (2013), 1460–1466.

Pizzi

N.J.

, Software quality prediction using fuzzy integration: A case study, Soft Computing12(1) (2008), 67–76.

Khramov

V.Y.

, and Besedin

P.N.

, Use of fuzzy situations in assessment of software quality, Telecommunications and Radio Engineering64(6) (2005), 455–464.

Shi

, and He

, Fuzzy software quality synthesis evaluation, Systems Engineering and Electronics24(12) (2002), 121–122.

Zhou

J.H.

, Wang

, and Yang

Z.K.

, et al. Research on software quality evaluation based on fuzzy method, Systems Engineering and Electronics26(7) (2004), 988–991.

Yang

Y.A.

, Synthetic evaluation method for software quality, Mini-Micro System21(3) (2000), 313–315.

Guo

Y.J.

, Comprehensive Evaluation Theory, Methods and Extensions, Science PressBeijing, China, 2012.

10.

Zhang

Y.M.

, Chi

G.T.

, and Xu

L.A.

, Comprehensive evaluation of human all-round development based on entropy method: Model and empirical study, Chinese Journal of Management6(5) (2009), 1047–1055.

11.

Ruan

, and Yang

, Software quality evaluation model based on weighted mutation rate correction incompletion G1 combination weights, Mathematical Problems in Engineering2014 (2014), Article ID 541292.

12.

Atanassov

, Intuitionistic fuzzy sets, Fuzzy Sets andSystems20 (1986), 87–96.

13.

Atanassov

, More on intuitionistic fuzzy sets, Fuzzy Sets and Systems33 (1989), 37–46.

14.

Zadeh

L.A.

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

15.

Hung

W.L.

, and Yang

M.S.

, On the J-divergence of intuitionistic fuzzy sets with its application to pattern recognition, Information Sciences178 (2008), 1641–1650.

16.

Z.S.

, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems15 (2007), 1179–1187.

17.

Z.S.

, Models for multiple attribute decision making with intuitionistic fuzzy information, International Journal of Uncertainty Fuzziness and Knowledge-Based Systems15 (2007), 285–297.

18.

Zhang

H.M.

, and Yu

L.Y.

, MADM method based on cross-entropy and extended TOPSIS with interval-valued intuitionistic fuzzy sets, Knowledge-Based Systems30(2) (2012), 115–120.

19.

Z.S.

, Some similarity measures of intuitionistic fuzzy sets and their applications to multiple attribute decision making, Fuzzy Optimization and Decision Making6 (2007), 109–121.

20.

Z.S.

, and Yager

R.R.

, Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group, Fuzzy Optimization and Decision Making8 (2009), 123–139.

21.

Wang

Z.J.

, Li

K.W.

, and Wang

W.Z.

, An approach to multiattribute decision making with interval-valued intuitionistic fuzzy assessments and incomplete weights, InformationSciences179 (2009), 3026–3040.

22.

Darvishi

, Fatemi

, and Faroughi

, Introduce a novel PCA method for intuitionistic fuzzy sets based on cross entropy, Applied Mathematics6(6) (2015), 990–995.

23.

X.Y.

, Interval-valued intuitionistic fuzzy continuous cross-entropy and its application in multi-attribute decision-making, Computer Engineering and Applications49(15) (2013), 234–237.

24.

Wei

G.W.

, Some induced geometric aggregation operators with intuitionistic fuzzy information and their application to group decision making, Applied Soft Computing10 (2010), 423–431.

25.

Wei

G.W.

, A novel efficient approach for interval-valued intuitionistic fuzzy multiple attribute decision making with incomplete weight information, Information-an International Interdisciplinary Journal14 (2011), 97–102.

26.

Liu

H.W.

, Wang

G.J.

, Multi-criteria decision-making methods based on intuitionistic fuzzy sets, European Journal of Operational Research179 (2007), 220–233.

27.

Mahapatra

G.S.

, Mitra

, Roy

T.K.

, Intuitionistic fuzzy multi-objective mathematical programming on reliability optimization model, International Journal of Fuzzy Systems12 (2010), 259–266.

28.

Zhao

X.F.

, Wei

G.W.

, Some intuitionistic fuzzy einstein hybrid aggregation operators and their application to multiple attribute decision making, Knowledge-Based Systems37 (2013), 472–479.

29.

Wei

G.W.

, Approaches to interval intuitionistic trapezoidal fuzzy multiple attribute decision making with incomplete weight information, International Journal of Fuzzy Systems17(3) (2015), 484–489.

30.

Chen

T.-Y.

, A comparative analysis of score functions for multiple criteria decision making in intuitionistic fuzzy settings, Information Sciences181(17) (2011), 3652–3676.

31.

Wei

G.W.

, Zhao

X.F.

, Lin

, Some hesitant interval-valued fuzzy aggregation operators and their applications to multiple attribute decision making, Knowledge-Based Systems46 (2013), 43–53.

32.

Z.S.

, Yager

R.R.

, Intuitionistic fuzzy bonferroni means, IEEE Transactions on Systems Man and Cybernetics Part B-Cybernetics41 (2011), 568–578.

33.

Wei

G.W.

, GRA method for multiple attribute decision making with incomplete weight information in intuitionistic fuzzy setting, Knowledge-Based Systems23 (2010), 243–247.

34.

Yager

, Some aspects of intuitionistic fuzzy sets, Fuzzy Optimization and Decision Making8 (2009), 67–90.

35.

Wei

G.W.

, Gray relational analysis method for intuitionistic fuzzy multiple attribute decision making, Expert Systems with Applications38 (2011), 11671–11677.

36.

Vlachos

I.K.

, Sergiadis

G.D.

, Intuitionistic fuzzy information— applications to pattern recognition, Pattern Recognition Letters28(2) (2007), 197–206.

37.

Wei

G.W.

, Wang

H.J.

, Lin

, Application of correlation coefficient to interval-valued intuitionistic fuzzy multiple attribute decision-making with incomplete weight information, Knowledge and Information Systems26 (2011), 337–349.

38.

Liu

, Yuan

X.H.

, Fuzzy number intuitionistic fuzzy set, Fuzzy Systems and Mathematics21(1) (2007), 88–91.

39.

Wang

X.F.

, Fuzzy number intuitionistic fuzzy arithmetic aggregation operators, International Journal of Fuzzy Systems10(2) (2008), 104–111.

40.

Wang

X.F.

, Fuzzy number intuitionistic fuzzy geometric aggregation operators and their application to decision making, Control and Decision23(6) (2008), 607–612.

41.

Lin

, Zhao

X.F.

, Wei

G.W.

, Fuzzy number intuitionistic fuzzy prioritized operators and their application to multiple attribute decision making, Journal of Intelligent and Fuzzy Systems24 (2013), 879–888.

42.

Hamachar

, Uber logische verknunpfungenn unssharfer Aussagen undderen Zugenhorige Bewertungsfunktione, in: Trappl , Klir , Riccardi (Eds.), Progress in Cybernatics and systems research, vol. 3, Hemisphere, Washington DC, (1978), pp. 276–288.

43.

Roychowdhury

, and Wang

B.H.

, On generalized Hamacher families of triangular operators, International Journal of Approximate Reasoning19 (1998), 419–439.

44.

Zhou

, and Chang

, Approach to multiple attribute decision making based on the Hamacher operation with fuzzy number intuitionistic fuzzy information and their application, Journal of Intelligent and Fuzzy Systems27(3) (2014), 1087–1094.

45.

Yager

R.R.

, The power average operator, IEEE Transactions on Systems, Man, and Cybernetics-Part A31(6) (2001), 724–731.

46.

Kim

S.H.

, and Ahn

B.S.

, Interactive group decision making procedure under incomplete information, European Journal of Operational Research116(3) (1999), 498–507.

47.

Joshi

, and Kumar

, Interval-valued intuitionistic hesitant fuzzy Choquet integral based TOPSIS method for multi-criteria group decision making, European Journal of Operational Research248(1) (2016), 183–191.

48.

Ran

L.-G.

, and Wei

, Uncertain prioritized operators and their application to multiple attribute group decision making, Technological and Economic Development of Economy21(1) (2015), 118–139.

49.

Wei

G.W.

, Wang

J.M.

, and Chen

, Potential optimality and robust optimality in multiattribute decision analysis with incomplete information: A comparative study, Decision Support Systems55(3) (2013), 679–684.

50.

Ouyang

, and Pedrycz

, A new model for intuitionistic fuzzy multi-attributes decision making, European Journal of Operational Research249(2) (2016), 677–682.

51.

Wang

, and Xu

, Multi-groups decision making using intuitionistic-valued hesitant fuzzy information, Int J Computational Intelligence Systems9(3) (2016), 468–482.

52.

, Additive intuitionistic fuzzy aggregation operators based on fuzzy measure, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems24(1) (2016), 1–12.

53.

Miguel

L.D.

, Bustince

, Fernández

, Induráin

, Kolesárová

, and Mesiar

, Construction of admissible linear orders for interval-valued Atanassov intuitionistic fuzzy sets with an application to decision making, Information Fusion27 (2016), 189–197.