Incentive-punitive risk function with interval valued intuitionistic fuzzy information for outsourced software project risk assessment

Abstract

By analyzing three parts of interval valued intuition fuzzy sets (IVIFS), we provide a novel framework constructing risk function with IVIFS information for risk analysis of outsourced software project. First, we introduced some risk factors of outsourced software project according to their hierarchical levels, and present a general risk function model. And then, some useful mathematical properties of this risk function are proved. Based on the mathematical properties of general risk function, all these presented specific risk functions are classified into four types: incentive function, punitive function, incentive-punitive function and equilibrium function. Especially, two kinds of incentive-punitive risk functions are focused on and a construction method for them is proposed. Finally, an application example for the risk assessment of outsourced software project illustrates the use of this decision making method. The simulation results show that these functions are effective in risk prediction.

Keywords

Interval valued intuitionistic fuzzy sets risk assessment incentive-punitive risk function equilibrium risk function outsourced software project

1 Introduction

Since Zadeh first introduced the concept of fuzzy set (FSs) [11], many extensions of the theory have been proposed, such as intuitionistic fuzzy set (IFS) [10], interval-valued intuitionistic fuzzy set (IVIFS) [9], interval-valued fuzzy set (IVFS) etc., which generalized FS theory. The introduction of IFS and IVIFS attracted many scholars to participate in the research. Some scholars applied them to multi attribute decision making and pattern recognition, and presented many order operators, similar operators, and distance measures. Some classical decision operators are gradually developed. According to Atanassov, two order operators derived from membership and non-membership degree were introduced. Hence, Chen & Tan [16] proposed a dominant order operator, Hong & Choi [5] introduced a clarity ordering operator,Xu & Xia [14, 28] presented a relative orderingoperator, Wei et al. [4] discussed how to construct order operator using distance, Zhang et al. [24 –26] introduced the concept of order maintaining, and applied it to the construction of order operators and distance measures.

Though many researchers studied IFS theory, few of them applied IFS to the study of risk assessment for software project development. Taking into account the fuzziness of Likert scale in outsourced software risk evaluation, it is necessary for us to introduce IFS method in this study.

In the research of software development risk, early scholars majorly focused on principles [2] and project risk index system [1–3 , 29]. Wallace, Keil, and Nidumolu et al. studied the constitute of output & decision attributes for software project risk, and analyzed the effect of condition attributes to decision attributes qualitatively [12 , 17]. Jones, Xu and Khoshgoftaar et al. studied the assessment and control of software risk [1, 29]. Hu, Zhang, and Nagi et al. utilized Bayesian networks to construct causal analysis framework for software project risk [21, 22], and predicted the risk by supervised combinational model [19, 20]. In fact, for the research of unsupervised model applied to software project risk prediction, only conventional clustering method and fuzzy neural network are presented [12, 30]. Based on the research result on risk index systems and IVIFS method, Zhang et al. presented a semi-supervised method and a dynamic method in 2016 [23, 27].

In this paper, we propose some risk functions of outsourced software project according to the analysis of the construction on IVIFS, and apply them to risk assessment of outsourced software development.

Firstly, we introduce the definition and some conventional order operators of IVIFS. Secondly, by analyzing the component of IVIFS, we present some risk functions with IVIFS information and their mathematical properties. Thirdly, the constructions of intermediate separation incentive-punitive and extreme separation incentive-punitive risk functions are introduced, respectively. Finally, we apply the conventional order operators and the presented risk functions to risk evaluation of outsourced software project development. The experimental results illustrate that all the proposed functions are valid methods.

2 Conventional weighted order operators

Definition 1. In universe X, an IVIFS A is defined by [9]: $A = {< x, M_{A} (x), N_{A} (x), H_{A} (x) > | x \in X}$ (1)

Where M_A (x) : X ⟶ [0, 1] , N_A (x) : X ⟶ [0, 1] , H_A (x) : X ⟶ [0, 1] with the condition: $\begin{matrix} \forall x \in X, M_{A} (x) = [t_{A}^{-} (x), t_{A}^{+} (x)] \subseteq [0, 1], \\ N_{A} (x) = [f_{A}^{-} (x), f_{A}^{+} (x)] \subseteq [0, 1], \\ t_{A}^{+} (x) + f_{A}^{+} (x) \leq 1 . \\ H_{A} (x) = [π_{A}^{-} (x), π_{A}^{+} (x)] \subseteq [0, 1], \\ π_{A}^{-} (x) = 1 - t_{A}^{+} (x) - f_{A}^{+} (x) \in [0, 1], \\ π_{A}^{+} (x) = 1 - t_{A}^{-} (x) - f_{A}^{-} (x) \in [0, 1] . \end{matrix}$

Where M_A (x), N_A (x) and H_A (x) are the interval of membership degree, non-membership degree and hesitant degree of x to A, respectively. According to the statistical background of IVIFS, it is well known that membership interval refers to the range of supporters, non-membership interval the scope of opponents, hesitation interval the range of the neutral.

Definition 2. For two IVIFSs A and B, we obtain: $\begin{matrix} A \subseteq B iff M_{A} (x) \leq M_{B} (x), N_{A} (x) \geq N_{B} (x), \\ M_{A} (x) \leq M_{B} (x) \Leftrightarrow t_{A}^{-} (x) \leq t_{B}^{-} (x), t_{A}^{+} (x) \leq t_{B}^{+} (x), \\ N_{A} (x) \geq N_{B} (x) \Leftrightarrow f_{A}^{-} (x) \geq f_{B}^{-} (x), f_{A}^{+} (x) \geq f_{B}^{+} (x) . \end{matrix}$

Obviously, the more supporters and the less opponents means the larger the membership degree and the smaller the non-membership degree for IVIFS, Thus, A being contained in B indicates that the supporters of A is less than that of B, and the opponents of A more than that of B.

From IVIFS, two weighted operators of membership and non-membership degree are defined: $R_{M} (A) = \sum_{x \in X} w_{A} (x) (t_{A}^{+} (x) + t_{A}^{-} (x)) .$ (2) $R_{NM} (A) = \sum_{x \in X} w_{A} (x) (f_{A}^{+} (x) + f_{A}^{-} (x)) .$ (3)

Chen & Tan [16] presented a weighted dominant ranking operator as follows:

$\begin{matrix} R_{CT} (A) \\ = \sum_{x \in X} w_{A} (x) ((t_{A}^{+} (x) - f_{A}^{+} (x)) \\ + (t_{A}^{-} (x) - f_{A}^{-} (x))) . \end{matrix}$ (4)

Hong & Choi [5] proposed a weighted clarity operator according to the sum of positive and negative comments:

$\begin{matrix} R_{HC} (A) \\ = \sum_{x \in X} w_{A} (x) ((t_{A}^{+} (x) + f_{A}^{+} (x)) \\ + (t_{A}^{-} (x) + f_{A}^{-} (x))) . \end{matrix}$ (5)

Where $μ_{A}^{-} (x), μ_{A}^{+} (x), ν_{A}^{-} (x)$ and $ν_{A}^{+} (x)$ are defined in Definition 1.

Xu presented the following formula (6) [14, 28]: $R_{Xu} (A) = \frac{m (A^{+}, A)}{m (A^{+}, A) + m (A^{-}, A)} .$ (6) $\begin{matrix} A^{+} & = & {< x, [max_{A \in Ω_{A}} (t_{A}^{-} (x)), max_{A \in Ω_{A}} (t_{A}^{+} (x))], \\ [min_{A \in Ω_{A}} (f_{A}^{-} (x)), min_{A \in Ω_{A}} (f_{A}^{+} (x))], | x \in X}, \\ A^{-} & = & {< x, [min_{A \in Ω_{A}} (t_{A}^{-} (x)), min_{A \in Ω_{A}} (t_{A}^{+} (x))], \\ [max_{A \in Ω_{A}} (f_{A}^{-} (x)), max_{A \in Ω_{A}} (f_{A}^{+} (x))], | x \in X} . \end{matrix}$

Xu proposed four operators from Equation (6) using four distance formulas.

3 Risk functions of IVIFS and their properties

In the following, we will construct some risk functions on outsourced software project with IVIFS information. We design a risk attribute framework.

We have collected more than 600 papers in this field since 1990s. According to our research in the past 10 years, we know that there are many risk factors of outsourced software project, involving three aspects: Contractee risk, contractor risk and the complexity of the project. Many risk factors has a significant positive correlation effect on the development of the outsourced software project, such as the standard level of project development, practical development experiment, communication skills and collaboration level between contractee and contractor, etc.

Next we will introduce some risk assessment methodology of outsourced software project with IVIFS information. Taking into account the specialty and particularity of the project development process of outsourced software, we utilize three first-level factors to analyze the risk of project development according to references [1–3 , 29]: contractor risks, customer risks and project complexity risks. And Table 1 contains all the second-level risk factors.

Table 1
Risk factors of outsourced software project analysis

Project Complexity Risks References Customer Risks References

1 Development Cost [29] 1 Client Team Collaboration [13 , 18]

2 Lines(KLOC) [1] 2 Top Management Support [13,15,18 , 13,15,18]

3 Number of Team Members [15] 3 Client Department Support [13 , 18]

4 Development Time [1] 4 Client Development Experiment [15, 18]

5 Technology Complexity [29] 5 Business Environment [18]

6 Fun Point [1] 6 Level of IT Application [13]

7 Real-time and Security [1] 7 Business Process [18]

8 Requirement Stability [3 , 13] Contractor Risks References

9 Number of Collaborators [3, 13] 1 Project Manager [13]

10 Schedule and Budget [13, 29] 2 Development Team [2 , 18]

11 Industry Experience [13, 18] 3 Plan &Control [2, 13]

4 Development &Test [8]

5 Engineering Support [6, 13]

Project Complexity Risks	References	Customer Risks	References
1 Development Cost	[29]	1 Client Team Collaboration	[13 , 18]
2 Lines(KLOC)	[1]	2 Top Management Support	[13,15,18 , 13,15,18]
3 Number of Team Members	[15]	3 Client Department Support	[13 , 18]
4 Development Time	[1]	4 Client Development Experiment	[15, 18]
5 Technology Complexity	[29]	5 Business Environment	[18]
6 Fun Point	[1]	6 Level of IT Application	[13]
7 Real-time and Security	[1]	7 Business Process	[18]
8 Requirement Stability	[3 , 13]	Contractor Risks	References
9 Number of Collaborators	[3, 13]	1 Project Manager	[13]
10 Schedule and Budget	[13, 29]	2 Development Team	[2 , 18]
11 Industry Experience	[13, 18]	3 Plan &Control	[2, 13]
		4 Development &Test	[8]
		5 Engineering Support	[6, 13]

According to previous research [19–22 , 30], we use 23 second-level risk factors with significant influence, 11 attributes of them belong to project complexity risks, 7 of them customs risks, and 5 them of them contractor risks. From reference [27], based on a survey data set focused on some small and medium companies undertaking software outsourced project from USA, Japan, and Southeast Asia, we analyzed all second-level risk factors by using supervised structural equation modeling, among them 14 secondary significant indexes (p < 0.05) with boldness are selected to form the most important attribute set.

For example, considering the project manager of contractor risk, we adopt 5-level Likert scale according to years of experience in software development, the number of software development, and cooperation and communication level. Obviously, for a project manager, the more experience and development ability he has, the less development risk will be. IF a manager is very excellent in development, his IVIFS values will be {< x, [1, 1] , [0, 0] , [0, 0] > |x ∈ X} over x, where x is project manager attribute. Thus, his development risk will reach the minimum 0. For the worst manager, his development risk will be the largest.

Target attribute is composed of 8 output attributes: Function, Performance, Information Quality, Maintainability, Satisfaction (Customer and User), Company Profits, Completion Degree in Time, Completion Degree in Budget [12, 17]. The options are Yes or No. A project is success project when 8 options are all Yes.

In present studies [19 , 30], we have proposed a specific risk analyzing framework for the project development of outsourced software, and made use of Bayesian networks to analyze its development risk.

Definition 3. Given A is an outsourced software project, and x ∈ X is an risk factor being taken into consideration, w is the weight vector of the attributes x. For each software project, its attribute value is shown by IVIFS as follows:

$\begin{matrix} A (x) & = & {< x, M_{A} (x), N_{A} (x), H_{A} (x) > | x \in X} \\ = & {< x, [t_{A}^{-} (x), t_{A}^{+} (x)], [f_{A}^{-} (x), f_{A}^{+} (x)], \\ [π_{A}^{-} (x), π_{A}^{+} (x)] > | x \in X} \\ = & {< x, [t_{A}^{-} (x), t_{A}^{+} (x)], [f_{A}^{-} (x), f_{A}^{+} (x)], \\ [1 - t_{A}^{+} (x) - f_{A}^{+} (x), 1 - t_{A}^{-} (x) - f_{A}^{-} (x)] \\ > & | x \in X} \end{matrix}$ (7)

Where the definition is the same with Definition 1. And M_A(x) indicates the support degree of the outstanding performance with regard to project A in factor x, while N_A(x) the opposition degree of the outstanding performance with respect to project A in factor x. For example: S = < x,, [0, 0], [0, 0] > | x ∈ X means M_S(x) = [1, 1], N_S(x) = [0, 0] and H_S(x) = [0, 0]; F = < x, [0, 0], [1, 1], [0, 0] > | x ∈ X means M_F(x) = [0, 0], N_F(x) = [1, 1] and H_F(x) = [0, 0]. Where S means that everyone thinks that all the influence factors perform excellently in this project, thus it represents a successful project (Membership degree intervals are all [1, 1]). On the contrary, F shows a failure project because its influence factors perform the worst (Non-Membership degree intervals are all [1, 1]). According to IVIFS, we provide a general risk function (8) for risk factor x ∈ X.

Definition 4.A is an outsourced software project, and x ∈ X is an risk factor. We note Risk (A (x)) to be a risk function of IVIFS A for risk attribute x ∈ X. Then a general risk is given by:

$\begin{matrix} Risk (A (x)) \\ = R^{*} (t_{A}^{-} (x), t_{A}^{+} (x), f_{A}^{-} (x), f_{A}^{+} (x), π_{A}^{-} (x), π_{A}^{+} (x)) \end{matrix}$ (8)

Where $t_{A}^{-} (x), t_{A}^{+} (x), f_{A}^{-} (x), f_{A}^{+} (x), π_{A}^{-} (x), π_{A}^{+} (x)$ are introduced in Definition 1.

However, it is hard to find a suitable elementary function and construct the related risk function from formula (7). Thus, we should seek some more concise expressions of risk functions. Considering the correspondence between variables, we define:

$\begin{matrix} A (x) & = & {< x, N_{A}^{max} (x), N_{A} (x), N_{A}^{mid} (x) > | x \in X} \\ = & {< x, [1 - t_{A}^{+} (x), 1 - t_{A}^{-} (x)], [f_{A}^{-} (x), f_{A}^{+} (x)], \\ [\frac{1 + f_{A}^{-} (x) - t_{A}^{-} (x)}{2}, \frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2}] \\ > & | x \in X} \end{matrix}$ (9)

Hence we obtain:

Definition 5. Assume that A is an outsourced software project, and x ∈ X is its risk factor. We define:

$\begin{matrix} Risk (A (x)) \\ = R^{*} (1 - t_{A}^{-} (x), 1 - t_{A}^{+} (x), f_{A}^{-} (x), f_{A}^{+} (x), \\ \frac{1 + f_{A}^{-} (x) - t_{A}^{-} (x)}{2}, \frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2}) \end{matrix}$ (10)

Theorem 1.Equation (8) is equivalent to Equation (10).

Proof. It is obvious that Equation (10) can be expressed by $t_{A}^{-} (x), t_{A}^{+} (x), f_{A}^{-} (x), f_{A}^{+} (x)$ . Thus, we only need to prove that Equation (8) can be expressed by: $\begin{matrix} 1 - t_{A}^{-} (x), 1 - t_{A}^{+} (x), f_{A}^{-} (x), f_{A}^{+} (x), \\ \frac{1 + f_{A}^{-} (x) - t_{A}^{-} (x)}{2}, \frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2} . \end{matrix}$

$t_{A}^{-} (x), t_{A}^{+} (x), f_{A}^{-} (x), f_{A}^{+} (x)$ are equivalent to $1 - t_{A}^{-} (x),$ $1 - t_{A}^{+} (x), f_{A}^{-} (x), f_{A}^{+} (x)$ . Thus, we only prove $π_{A}^{-} (x)$ and $π_{A}^{+} (x)$ can be expressed by $1 - t_{A}^{-} (x), 1 - t_{A}^{+} (x),$ $f_{A}^{-} (x), f_{A}^{+} (x) .$ According to $π_{A}^{-} (x) = 1 - t_{A}^{+} (x) - f_{A}^{+} (x),$ $π_{A}^{+} (x) = 1 - t_{A}^{-} (x) - f_{A}^{-} (x)$ , we know that $π_{A}^{-} (x)$ and $π_{A}^{+} (x)$ can also be represented by $1 - t_{A}^{-} (x), 1 - t_{A}^{+} (x),$ $f_{A}^{-} (x), f_{A}^{+} (x)$ easily. Therefore, Equation (8) is equivalent to Equation (10).

Theorem 2. For two IVIFSs A & B, we have:

If R^* (a₁, a₂, a₃, a₄, a₅, a₆) satisfy: For each a_k, $\frac{\partial R^{*}}{\partial a_{k}} \geq 0, (k = 1, 2, 3, 4, 5, 6),$ then we obtain: $A \subseteq B \Rightarrow Risk (A (x)) \geq Risk (B (x)) .$

Theorem 2 shows that monotone increasing function can maintain the order of IVIFS, which means that if A and B are two IVIFSs and A ⊆ B, and the risk function is a monotone increasing function, then we conclude that the risk of A is more than that of B.

Proof. For each a_k, $\frac{\partial R^{*}}{\partial a_{k}} \geq 0, (k = 1, 2, 3, 4, 5, 6),$ if a_k ≤ b_k (k = 1, 2, 3, 4, 5, 6), then we have:

R^* (a₁, a₂, a₃, a₄, a₅, a₆) ≤ R^* (b₁, b₂, b₃, b₄, b₅, b₆) . For two IVIFSs A & B, and A ⊆ B is equivalent to M_A (x) ≤ M_B (x), N_A (x) ≥ N_B (x), then $\begin{matrix} \to {\begin{matrix} t_{A}^{-} (x) \leq t_{B}^{-} (x) \\ t_{A}^{+} (x) \leq t_{B}^{+} (x) \\ f_{A}^{-} (x) \geq f_{B}^{-} (x) \\ f_{A}^{+} (x) \geq f_{B}^{+} (x) \end{matrix} \\ \to {\begin{matrix} 1 - t_{A}^{-} (x) \geq 1 - t_{B}^{-} (x) \\ 1 - t_{A}^{+} (x) \geq 1 - t_{B}^{+} (x) \\ \frac{1 + f_{A}^{-} (x) - t_{A}^{-} (x)}{2} \geq \frac{1 + f_{B}^{-} (x) - t_{B}^{-} (x)}{2} \\ \frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2} \geq \frac{1 + f_{B}^{+} (x) - t_{B}^{+} (x)}{2} \end{matrix} \\ \to R^{*} (1 - t_{A}^{-} (x), 1 - t_{A}^{+} (x), f_{A}^{-} (x), f_{A}^{+} (x), \\ \frac{1 + f_{A}^{-} (x) - t_{A}^{-} (x)}{2}, \frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2}) \\ \geq R^{*} (1 - t_{B}^{-} (x), 1 - t_{B}^{+} (x), f_{B}^{-} (x), f_{B}^{+} (x), \\ \frac{1 + f_{B}^{-} (x) - t_{B}^{-} (x)}{2}, \frac{1 + f_{B}^{+} (x) - t_{B}^{+} (x)}{2}) \\ \to Risk (A (x)) \geq Risk (B (x)) . \end{matrix}$

Theorem 1 demonstrates that Formula (8) is equivalent to Formula (10), which shows that Equation (10) can be applied to construct risk function.

Theorem 2 proves that each monotonically increasing function in [0, 1]⁶ can be used as a risk assessment model.

Theorem 3. For two IVIFSs A & B, R^* : [0, 1] ⁶ ⟶ RS is a monotonically increasing function in [0,1]⁶, then $\begin{matrix} R^{*} (1 - μ_{A}^{-} (x), 1 - μ_{A}^{+} (x), ν_{A}^{-} (x), ν_{A}^{+} (x), \\ \frac{1 + ν_{A}^{-} (x) - μ_{A}^{-} (x)}{2}, \frac{1 + ν_{A}^{+} (x) - μ_{A}^{+} (x)}{2}) \end{matrix}$ is a suitable risk function satisfying A ⊆ B ⇒ R^* (A (x)) ≥ R^* (B (x)). Where risk value set RS is a real number set.

Let

$\begin{matrix} R^{*} (A (x)) \\ = \sum_{x \in X} ω (x) {R (1 - t_{A}^{-} (x)) \\ + R (1 - t_{A}^{+} (x)) + R (f_{A}^{-} (x)) + R (f_{A}^{+} (x)) \\ + R (\frac{1 + f_{A}^{-} (x) - t_{A}^{-} (x)}{2}) \\ + R (\frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2})} \end{matrix}$ (11)

Where R(a_k), the component function of R^* (A (x)), is an increasing function in the domain [0, 1].

According to Theorem 2, each monotonically increasing function in [0, 1]⁶ is with the condition being a risk function. If R (a_k) is a monotonically increasing function in [0, 1], then R^* (A (x)) is a risk function. If R (a_k) is a linear function, then R (a_k) changes evenly followed with a_k. Thus it is an equilibrium risk-function. If R (a_k) is a monotonically increasing convex function, then it is easy to infer that when a_k approaches boundary value 0, the risk function value is sharply reduced to 0 and the possibility of success of the project greatly increased. Hence, we define it incentive risk-function. Similarly an increasing concave function means a punitive risk-function.

Definition 6. For formula (11), if R(a_k) is a monotonically increasing convex function over [0,1], we call R^* (A (x)) incentive risk-function; if R(a_k) is a monotonically increasing concave function over [0,1], we call it punitive risk-function; and if R(a_k) is a monotonically increasing linear function over [0,1], we call it equilibrium risk-function.

Theorem 4. For formula (11), if $\frac{d^{2} R (a_{k})}{\partial a_{k}^{2}} < 0$ and $\frac{dR (a_{k})}{{da}_{k}} \geq 0,$ R(a_k) is an incentive risk-function; if $\frac{d^{2} R (a_{k})}{{da}_{k}^{2}} > 0 & \frac{dR (a_{k})}{{da}_{k}} \geq 0,$ R(a_k) is a punitive risk-function; and if $\frac{d^{2} R (a_{k})}{{da}_{k}^{2}} = 0 & \frac{dR (a_{k})}{{da}_{k}} \geq 0,$ R(a_k) is an equilibrium risk-function.

Inference 1. For formula (11), if $\frac{\partial^{2} R^{*} (a_{k})}{\partial a_{k}^{2}} < 0$ , R^* (A (x)) is an intermediate separation incentive risk function; if $\frac{\partial^{2} R^{*} (a_{k})}{\partial a_{k}^{2}} > 0$ , R^* (A (x)) is a punitive risk function; and if $\frac{\partial^{2} R^{*} (a_{k})}{\partial a_{k}^{2}} \equiv 0$ , R^* (A (x)) is an equilibrium risk-function.

Definition 7. For formula (11), R(a_k) is a monotonically increasing function over [0,1], which means $\frac{dR (a_{k})}{{da}_{k}} \geq 0,$ . There is a threshold value a, if R (a_k) satisfies: When $0 \leq a_{k} \leq a, \frac{d^{2} R (a_{k})}{{da}_{k}^{2}} > 0;$ when $a \leq a_{k} \leq 1, \frac{d^{2} R (a_{k})}{{da}_{k}^{2}} < 0;$ then R^* (A (x)) is an intermediate separation incentive-punitive risk function. On the contrary, if R (a_k) satisfies:

When 0 ≤ a_k ≤ a, $\frac{d^{2} R (a_{k})}{{da}_{k}^{2}} < 0;$ when a ≤ a_k≤1, $\frac{d^{2} R (a_{k})}{{da}_{k}^{2}} > 0;$ then R^* (A (x)) is an extreme separation incentive-punitive risk function.

Figure 1a shows an example of intermediate-separation incentive-punitive risk function. For this function, when the risk variable a_k approaches boundary value (0 or 1), the risk function value changes slowly, and the discrimination degree between every two IVIFSs is small. When the risk variable approaches the middle 0.5, the risk function value changes quickly and the discrimination degree between every two IVIFSs is large. On the contrary, Fig. 1b illustrates an example of extreme-separation incentive-punitive risk function. It means that when the risk variable a_k approaches boundary value (0 or 1), the risk function value changes quickly and the discrimination degree between every two IVIFSs is large. When the risk variable approaches the middle 0.5, the risk function value changes slowly, and the discrimination degree between every two IVIFSs is small. From Fig. 1, the former is suitable for stimulating the separation between high-risk and low-risk from middle part, while the latter mainly separate the high-risk and low-risk in the two endpoints 0 and 1.

Fig.1

Classification of incentive-punitive risk function. (a) Intermediate separation function. (b) Extreme separation function.

Based on Equation (11), we introduce a component function (12) of the incentive-punitive risk function R^* (A (x)) as follows. $R (a_{k}) = β_{3} a_{k}^{3} + β_{2} a_{k}^{2} + β_{1} a_{k} + β_{0}$ (12)

According to Definition 8, $\frac{dR (a_{k})}{{da}_{k}} \geq 0,$ and then we obtain: $\frac{dR (a_{k})}{{da}_{k}} = 3 β_{3} a_{k}^{2} + 2 β_{2} a_{k} + β_{1} \geq 0$ (13)

Assume that f (a_k) satisfies: $R (0) = 0, R (0.5) = 0.5, R (1) = 1 .$

Hence we have: $R (a_{k}) = (2 β - 2) a_{k}^{3} + (3 - 3 β) a_{k}^{2} + β a_{k}$ (14)

Where $\begin{matrix} {\begin{matrix} R (0) = 0, \\ R (0.5) = 0.5, \\ R (1) = 1, \end{matrix} \Rightarrow {\begin{matrix} β_{0} = 0, \\ β_{3} + 2 β_{2} + 4 β_{1} = 4, \\ β_{3} + β_{2} + β_{1} = 1, \end{matrix} \\ \Rightarrow {\begin{matrix} β_{0} = 0, \\ β_{2} = 3 - 3 β_{1}, \\ β_{3} = 2 β_{1} - 2, \end{matrix} \underset{Let β_{1} = β}{\Rightarrow} {\begin{matrix} β_{0} = 0, β_{1} = β, \\ β_{2} = 3 - 3 β, \\ β_{3} = 2 β - 2, \end{matrix} \\ \Rightarrow R (a_{k}) = (2 β - 2) a_{k}^{3} + (3 - 3 β) a_{k}^{2} + β a_{k} . \end{matrix}$

Lemma 1. Necessary Condition. From definition 8 and formula (13), a necessary condition of R (a_k) is: for each a_k∈ [0, 1], we have $\frac{dR (a_{k})}{{da}_{k}} = (6 β - 6) a_{k}^{2} + (6 - 6 β) a_{k} + β \geq 0$ (15)

From equation (14), we have

$\begin{matrix} \frac{d^{2} R (a_{k})}{{da}_{k}^{2}} & = & 12 (β - 1) a_{k} + 6 (1 - β) \\ = & 6 (2 a_{k} - 1) (β - 1) \end{matrix}$ (16)

Thus, we draw a conclusion under the precondition of Lemma 1:

When β < 1, if 0 ≤ a_k < 0.5 then $\frac{d^{2} R (a_{k})}{{da}_{k}^{2}} > 0$ , and if 0.5 < a_k ≤ 1 then $\frac{d^{2} R (a_{k})}{{da}_{k}^{2}} < 0$ . It denotes an intermediate separation risk function, as is shown in Fig. 1a.

When β > 1, if 0 ≤ a_k < 0.5 then $\frac{d^{2} R (a_{k})}{{da}_{k}^{2}} < 0$ , and if 0.5 < a_k ≤ 1 then $\frac{d^{2} R (a_{k})}{{da}_{k}^{2}} > 0$ . It denotes an extreme separation risk function, as is shown in Fig. 1b.

Therefore, we know that if the components function of R (a_k) satisfies Lemma 1:

$(6 β - 6) a_{k}^{2} + (6 - 6 β) a_{k} + β \geq 0,$ then R^* (A (x)) is an incentive-punitive risk function.

Lemma 2. Sufficient Condition. When 0 ≤ β < 1, R^* (A (x)) is an intermediate separation incentive-punitive risk function; when 1 < β ≤ 3, R^* (A (x)) is an extreme separation incentive-punitive risk function. when β = 1, R^* (A (x)) is an equilibrium risk-function.

Obviously, when β = 1, R (a_k) = a_k, R^* (A (x)) is a linear function of the equilibrium change. The other two cases have been analyzed from Lemma 1 above.

From Fig. 1, the following properties are obvious. Compared with traditional linear risk function, the risk function value is significantly changed in the middle of the attribute value for intermediate separation incentive-punitive risk function (in Fig. 1a), and the deviation degree of risk will be larger. For conditional attributes, the difference of risk values based on most of good performance projects and poor performance projects are widen and significantly distinguished, hence the variance, standard deviation, and range of intermediate separation risk function will be large than that of linear risk function.

For the extreme separation incentive-punitive risk function (in Fig. 1b), its properties are different from the properties of intermediate separation risk function and linear risk function. According to Fig. 1b, only the extreme value will be increased or decreased sharply, which means that only the difference between very good performance projects and the very terrible performance projects are widen and significantly distinguished by the risk values, while most risk values of all projects may be changed slowly. Thus, the variance, standard deviation, and range of intermediate separation risk function will be smaller than that of linear risk function and intermediate separation incentive-punitive risk function.

If β = 1, we have $\frac{\partial^{2} R (a_{k})}{\partial a_{k}^{2}} = 0$ and $\frac{\partial^{2} R^{*} (a_{k})}{\partial a_{k}^{2}} = 0$ , which means a liner risk-function for risk attribute a_k and we call it an equilibrium risk-function.

Next we introduce some component functions meeting the needs of Lemma 2 and construct their incentive-punitive risk functions $R_{β}^{*} (A (x))$ , we adopt β = 0, 0.5, 1, 2, 3:

$\begin{matrix} R_{β}^{*} (A (x)) \\ = \sum_{x \in X} ω (x) {R_{β} (1 - t_{A}^{-} (x)) \\ + R_{β} (1 - t_{A}^{+} (x)) + R_{β} (f_{A}^{-} (x)) + R_{β} (f_{A}^{+} (x)) \\ + R_{β} (\frac{1 + f_{A}^{-} (x) - t_{A}^{-} (x)}{2}) \\ + R_{β} (\frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2})} \end{matrix}$ (17)

Intermediate separation incentive-punitive risk function: $R_{0} (a_{k}) = - 2 a_{k}^{3} + 3 a_{k}^{2}, (β = 0)$ (18) $R_{0.5} (a_{k}) = - a_{k}^{3} + 1.5 a_{k}^{2} + 0.5 a_{k}, (β = 0.5)$ (19)

Extreme separation incentive-punitive risk function: $R_{2} (a_{k}) = 2 a_{k}^{3} - 3 a_{k}^{2} + 2 a_{k}, (β = 2)$ (20) $R_{3} (a_{k}) = 4 a_{k}^{3} - 6 a_{k}^{2} + 3 a_{k}, (β = 3)$ (21)

Equilibrium risk-function: $R_{1} (a_{k}) = a_{k}, (β = 1)$ (22)

Obviously, for two IVIFSs A & B, if A ⊆ B, then $R_{β}^{*} (A (x)) \leq R_{β}^{*} (B (x))$ .

4 Application example

Example 1. A project manager looks forward to evaluating the project development risk of outsourced software. If A_i (i = 1, 2, 3, 4, 5) is software project. Consider three attributes: contractor risks (x₁), customer risks (x₂), and project complexity risks (x₃). And w = (0.5, 0.3, 0.2) ^T denotes the weight matrix of three attributes x_j (j = 1, 2, 3). A_i (i = 1, 2, 3, 4, 5) is given by the following IVIFS modes: $\begin{matrix} A_{1} & = & {< x_{1}, [0.6, 0.7], [0, 0.1] >, \\ < x_{2}, [0.1, 0.2], [0.3, 0.4] >, \\ < x_{3}, [0.5, 0.6], [0.2, 0.3] >}, \\ A_{2} & = & {< x_{1}, [0.4, 0.5], [0.1, 0.2] >, \\ < x_{2}, [0.3, 0.4], [0.1, 0.2] >, \\ < x_{3}, [0.7, 0.8], [0, 0.1] >}, \\ A_{3} & = & {< x_{1}, [0.5, 0.6], [0.1, 0.2] >, \\ < x_{2}, [0.4, 0.5], [0.3, 0.4] >, \\ < x_{3}, [0.8, 0.9], [0, 0] >}, \\ A_{4} & = & {< x_{1}, [0.7, 0.8], [0, 0.1] >, \\ < x_{2}, [0.2, 0.3], [0.4, 0.5] >, \\ < x_{3}, [0.6, 0.7], [0.1, 0.2] >}, \\ A_{5} & = & {< x_{1}, [0.6, 0.7], [0, 0] >, \\ < x_{2}, [0.7, 0.8], [0.1, 0.2] >, \\ < x_{3}, [0, 0.1], [0.5, 0.6] >} . \end{matrix}$

According to formula (9), we get $\begin{matrix} A_{1} & = & {< x_{1}, [0.3, 0.4], [0, 0.1], [0.2, 0.2] >, \\ < x_{2}, [0.8, 0.9], [0.3, 0.4], [0.6, 0.6] >, \\ < x_{3}, [0.4, 0.5], [0.2, 0.3], [0.35, 0.35] >}, \\ A_{2} & = & {< x_{1}, [0.5, 0.6], [0.1, 0.2], [0.35, 0.35] >, \\ < x_{2}, [0.6, 0.7], [0.1, 0.2], [0.4, 0.4] >, \\ < x_{3}, [0.2, 0.3], [0, 0.1], [0.15, 0.15] >}, \\ A_{3} & = & {< x_{1}, [0.4, 0.5], [0.1, 0.2], [0.3, 0.3] >, \\ < x_{2}, [0.5, 0.6], [0.3, 0.4], [0.45, 0.45] >, \\ < x_{3}, [0.1, 0.2], [0, 0], [0.05, 0.1] >}, \\ A_{4} & = & {< x_{1}, [0.2, 0.3], [0, 0.1], [0.15, 0.15] >, \\ < x_{2}, [0.7, 0.8], [0.4, 0.5], [0.6, 0.6] >, \\ < x_{3}, [0.3, 0.4], [0.1, 0.2], [0.25, 0.25] >}, \\ A_{5} & = & {< x_{1}, [0.3, 0.4], [0, 0], [0.15, 0.2] >, \\ < x_{2}, [0.2, 0.3], [0.1, 0.2], [0.2, 0.2] >, \\ < x_{3}, [0.9, 1.0], [0.5, 0.6], [0.75, 0.75] >} . \end{matrix}$

From formulas (17, 22), we obtain the results below: $\begin{matrix} R_{1}^{*} (A_{1} (x)) \\ = \sum_{x \in X} ω (x) {R_{1} (1 - t_{A}^{-} (x)) + R_{1} (1 - t_{A}^{+} (x)) \\ + R_{1} (f_{A}^{-} (x)) + R_{1} (f_{A}^{+} (x)) \\ + R_{1} (\frac{1 + f_{A}^{-} (x) - t_{A}^{-} (x)}{2}) \\ + R_{1} (\frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2})} \\ = \sum_{x \in X} ω (x) {(1 - t_{A}^{-} (x)) + (1 - t_{A}^{+} (x)) \\ + (f_{A}^{-} (x)) + (f_{A}^{+} (x)) + (\frac{1 + f_{A}^{-} (x) - t_{A}^{-} (x)}{2}) \\ + (\frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2})} \\ = [0.5 \times (0.4 + 0.3 + 0 + 0.1 + 0.2 + 0.2) \\ + 0.3 \times (0.9 + 0.8 + 0.3 + 0.4 + 0.6 + 0.6) \\ + 0.2 \times (0.5 + 0.4 + 0.2 + 0.3 + 0.35 + 0.35)] \\ = 2.1 . \end{matrix}$

From formulas (17, 18), we get the result as follows. Similarly, we get the following Table 2.

Table 2
Results based on risk functions and conventional order operators of IVIFS

Operators A₁ A₂ A₃ A₄ A₅ Decision-making

R₁^* 2.1 1.95 1.8 1.83 1.785 A₅ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₁

R₀^* 1.8687 1.6961 1.5114 1.5857 1.5259 A₃ ≻ A₅ ≻ A₄ ≻ A₂ ≻ A₁

R_0.5^* 1.9844 1.8230 1.6557 1.7078 1.6554 A₅ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₁

R₂^* 2.3313 2.2040 2.0886 2.0744 2.0441 A₅ ≻ A₄ ≻ A₃ ≻ A₂ ≻ A₁

R₃^* 2.5626 2.4579 2.3772 2.3187 2.3033 A₅ ≻ A₄ ≻ A₃ ≻ A₂ ≻ A₁

R_M 0.480 0.480 0.580 0.580 0.560 A₃ = A₄ ≻ A₅ ≻ A₂ = A₁

R_NM 0.180 0.130 0.180 0.190 0.155 A₂ = A₅ ≻ A₃ ≻ A₁ ≻ A₄

R_CT 0.300 0.350 0.400 0.390 0.405 A₅ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₁

R_HC 0.660 0.610 0.760 0.770 0.715 A₄ ≻ A₃ ≻ A₅ ≻ A₁ ≻ A₂

Operators	A₁	A₂	A₃	A₄	A₅	Decision-making
R₁^*	2.1	1.95	1.8	1.83	1.785	A₅ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₁
R₀^*	1.8687	1.6961	1.5114	1.5857	1.5259	A₃ ≻ A₅ ≻ A₄ ≻ A₂ ≻ A₁
R_0.5^*	1.9844	1.8230	1.6557	1.7078	1.6554	A₅ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₁
R₂^*	2.3313	2.2040	2.0886	2.0744	2.0441	A₅ ≻ A₄ ≻ A₃ ≻ A₂ ≻ A₁
R₃^*	2.5626	2.4579	2.3772	2.3187	2.3033	A₅ ≻ A₄ ≻ A₃ ≻ A₂ ≻ A₁
R_M	0.480	0.480	0.580	0.580	0.560	A₃ = A₄ ≻ A₅ ≻ A₂ = A₁
R_NM	0.180	0.130	0.180	0.190	0.155	A₂ = A₅ ≻ A₃ ≻ A₁ ≻ A₄
R_CT	0.300	0.350	0.400	0.390	0.405	A₅ ≻ A₃ ≻ A₄ ≻ A₂ ≻ A₁
R_HC	0.660	0.610	0.760	0.770	0.715	A₄ ≻ A₃ ≻ A₅ ≻ A₁ ≻ A₂

5 Experimental result analysis and discussion

Experimental result analysis.

In Table 2, R_M represents the weighted membership and R_CT means a superiority sorting function with the difference of Membership minus non-membership. Therefore, for R_M & R_CT, the larger the value of them, the smaller the software project risk and the higher the development success rate. On the contrary, for each i > 0, $R_{β}^{*} (A_{i} (x))$ is risk function, which means that it possess exactly the opposite properties with R_M & R_CT: the smaller their values, the smaller the software project risk and the higher the development success rate. R_NM is composed by weighted non-membership degree, and it means that R_NM is also a kind of risk function as $R_{β}^{*} (A_{i} (x))$ .

According to Table 2, we obtain: $\begin{matrix} R_{0}^{*} (A_{1} (x)) & = & \sum_{x \in X} ω (x) {R_{0} (1 - t_{A}^{-} (x)) + R_{0} (1 - t_{A}^{+} (x)) \\ + R_{0} (f_{A}^{-} (x)) + R_{0} (f_{A}^{+} (x)) + R_{0} (\frac{1 + f_{A}^{-} (x) - t_{A}^{-} (x)}{2}) + R_{0} (\frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2})} \\ = & \sum_{x \in X} ω (x) {- 2 (1 - t_{A}^{-} (x))^{3} + 3 (1 - t_{A}^{-} (x))^{2} - 2 (1 - t_{A}^{+} (x))^{3} + 3 (1 - t_{A}^{+} (x))^{2} \\ - 2 (f_{A}^{-} (x))^{3} + 3 (f_{A}^{-} (x))^{2} - 2 (f_{A}^{+} (x))^{3} \\ + 3 (f_{A}^{+} (x))^{2} - 2 {(\frac{1 + f_{A}^{-} (x) - μ_{A}^{-} (x)}{2})}^{3} + 3 {(\frac{1 + f_{A}^{-} (x) - μ_{A}^{-} (x)}{2})}^{2} \\ - 2 {(\frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2})}^{3} + 3 {(\frac{1 + f_{A}^{+} (x) - t_{A}^{+} (x)}{2})}^{2}} \\ = & [0.5 \times (- 2 \times 0 . 4^{3} + 3 \times 0 . 4^{2} - 2 \times 0 . 3^{3} + 3 \times 0 . 3^{2} - 2 \times 0^{3} + 3 \times 0^{2} - 2 \times 0 . 1^{3} \\ + 3 \times 0 . 1^{2} - 2 \times 0 . 2^{3} + 3 \times 0 . 2^{2} - 2 \times 0 . 2^{3} + 3 \times 0 . 2^{2}) \\ + 0.3 \times (- 2 \times 0 . 9^{3} + 3 \times 0 . 9^{2} - 2 \times 0 . 8^{3} + 3 \times 0 . 8^{2} - 2 \times 0 . 3^{3} + 3 \times 0 . 3^{2} \\ - 2 \times 0 . 4^{2} + 3 \times 0 . 4^{2} - 2 \times 0 . 6^{3} + 3 \times 0 . 6^{2} - 2 \times 0 . 6^{3} + 3 \times 0 . 6^{2}) \\ + 0.2 \times (- 2 \times 0 . 5^{3} + 3 \times 0 . 5^{2} - 2 \times 0 . 4^{3} + 3 \times 0 . 4^{2} - 2 \times 0 . 2^{3} + 3 \times 0 . 2^{2} \\ - 2 \times 0 . 3^{3} + 3 \times 0 . 3^{2} - 2 \times 0 . 35^{3} + 3 \times 0 . 35^{2} - 2 \times 0 . 35^{3} + 3 \times 0 . 35^{2})] = 1.8687 . \\ R_{M} (A_{3}) & = & R_{M} (A_{4}) > R_{M} (A_{5}) > R_{M} (A_{1}) = R_{M} (A_{2}), R_{NM} (A_{2}) < R_{NM} (A_{5}) < R_{NM} (A_{1}) \\ = & R_{NM} (A_{3}) < R_{NM} (A_{4}) . \end{matrix}$

Thus we get A₂ ≻ A₁, A₅ ≻ A₁, A₃ ≻ A₁, A₃ ≻ A₄ .

For example, from the membership degree R_M (A₅) > R_M (A₁) = R_M (A₂) and the non-membership degree R_NM (A₂) < R_NM (A₅) < R_NM (A₁), we obtain A₅ ≻ A₁, A₂ ≻ A₁. Similarly, we have A₃ ≻ A₁, A₃ ≻ A₄. Hence, the optimal decision-making is from set A₂, A₃, A₅.

From Table 2, R_CT, R_NM, R₁^*, R₀^* and R_0.5^* satisfy A₂ ≻ A₁, A₃ ≻ A₁, A₅ ≻ A₁, A₃ ≻ A₄ . While R_M, R_HC don’t satisfy A₃ ≻ A₄ and A₂ ≻ A₁, R₂^* and R₃^* don’t satisfy A₃ ≻ A₄.

Discussion.

From Example 1, for contractor risks (x₁) and customer risks (x₂), A₅ is outstanding, while it performs the worst in project complexity risks (x₃).

A₃ and A₄ show their advantages in x₁ and x₃, but A₄ is terrible in x₂. In general, A₃ performs well in all three respects, and the overall performance of A₂, A₃. and A₄ are homogeneous in all attributes.

However, A₁ and A₅ show extreme performance on their attributes.

Obviously, A₃ and A₅ are closer to successful projects while A₁ failure project.

Considering $R_{β}^{*} (A_{i} (x))$ , Let: $\begin{matrix} S & = & {< x_{1}, [1, 1], [0, 0] >, < x_{2}, [1, 1], [0, 0] >, \\ < x_{3}, [1, 1], [0, 0] >}, \\ M & = & {< x_{1}, [0.5, 0.5], [0.5, 0.5] >, \\ < x_{2}, [0.5, 0.5], [0.5, 0.5] >, \\ < x_{3}, [0.5, 0.5], [0.5, 0.5] >}, \\ F & = & {< x_{1}, [0, 0], [1, 1] >, \\ < x_{2}, [0, 0], [1, 1] >, < x_{3}, [0, 0], [1, 1] >} . \end{matrix}$

Where T means all the membership degree values of attributes x₁, x₂, x₃ are 1 and all the non-membership degree values of attributes x₁, x₂, x₃ are 0. Thus the risk of project T is 0 and T can be defined as a successful project. On the contrary, the membership degree values of attributes x₁, x₂, x₃ for F are 0 and its non-membership degree values 1. Therefore, F is regarded as a failure project and its risk will be the maximum 6. M represents middle level of risk. By calculating $R_{β}^{*} (S), R_{β}^{*} (F), R_{β}^{*} (M)$ , we obtain Table 3.

Table 3
Risk level and deviation of several risk functions with IVIFS information

Prediction Accuracy S M F Variance Standard Deviation Range

R₁^* 0 3 6 0.0176 0.1327 0.315

R₀^* 0 3 6 0.02198 0.1483 0.3573

R_0.5^* 0 3 6 0.01968 0.1403 0.3289

R₂^* 0 3 6 0.01412 0.1188 0.2872

R₃^* 0 3 6 0.01155 0.1075 0.2594

Prediction Accuracy	M	F	Variance	Standard Deviation	Range
R₁^*	3	6	0.0176	0.1327	0.315
R₀^*	3	6	0.02198	0.1483	0.3573
R_0.5^*	3	6	0.01968	0.1403	0.3289
R₂^*	3	6	0.01412	0.1188	0.2872
R₃^*	3	6	0.01155	0.1075	0.2594

According to A_i(x), most of the attribute values $\begin{array}{l} A_{i} (x_{k}) \\ = {< x, N_{A_{i}}^{\max} (x_{k}), N_{A_{i}} (x_{k}), N_{A_{i}}^{m i d} (x_{k}) > |} \\ {x \in X, i \in {1, 2, 3, 4, 5}, k \in {1, 2, 3} \end{array}$ applied to $R_{β}^{*} (A_{i} (x_{k}))$ be less than 0.5, therefore the values of the extreme separation risk functions will be larger than that of the equilibrium risk function, and the values calculated by equilibrium risk function are larger than that by the extreme separation risk functions. For example, when 1 < β ≤ 3, all the risk values will be increased, and the higher the degree of non-membership is, the closer to failure the project is and the faster growth the project risk will be.

Similarly, When 0 ≤ β < 1, all the risk values will be decreased, and the higher the degree of membership is, indicating that the closer to success the project is and the rapid decline the project risk will be.

When β = 1, the risk values is changed linearly.

Discussion on the risk value deviations of all the risk functions in Table 3. From Table 3, we know that the deviation of the extreme separation risk functions will be less than that of the equilibrium risk function, and the values calculated by the equilibrium risk function are smaller than that by the extreme separation risk functions, which is the same as the properties deduced in theory.

We draw a conclusion: For $R_{β}^{*} (A_{i} (x)), β \in [0, 3],$ if 0 ≤ β_j < 1 < β_l ≤ 3, $R_{β_{j}}^{*} (A_{i} (x)) < R_{1}^{*} (A_{i} (x)) < R_{β_{l}}^{*} (A_{i} (x))$ , the deviations of $R_{β_{l}}^{*} (A_{i} (x))$ are less than that of $R_{1}^{*} (A_{i} (x))$ , and the deviations of $R_{1}^{*} (A_{i} (x))$ are less than that of $R_{β_{j}}^{*} (A_{i} (x))$ .

6 Outsourced software risk assessment

In our survey, we collect 260 valid and complete individuals, 191 of them are success and 69 failure. The average accuracy of 10 tests is shown in Table 4. The algorithm steps are shown below:

Table 4
Average accuracy of IVIFS

Risk Prediction Ranking Prediction

Operator Accuracy Function Accuracy

R₁^* 0.831 R_M 0.804

R₀^* 0.827 R_NM 0.796

R_0.5^* 0.819 R_CT 0.831

R₂^* 0.823 R_HC 0.723

R₃^* 0.807

Risk	Prediction	Ranking	Prediction
R₁^*	0.831	R_M	0.804
R₀^*	0.827	R_NM	0.796
R_0.5^*	0.819	R_CT	0.831
R₂^*	0.823	R_HC	0.723
R₃^*	0.807

Step 1. Sampling.

In the experiment, we sample 200 individuals to be training sample and the rest testing sample.

Step 2. Fuzzification and weights determination.

The data set consists of 14 significant attributes with 5 level scale, we use triangle module to define membership, non-membership, and hesitation interval. For all risk factors, we use the equal weights to construct risk function.

Step 3. The threshold value determination.

Considering that the success rate of project is 26.5%, 53 individuals with the lowest risk are regarded as success projects in training example.

Step 4. Testing.

According to the threshold value, calculate the risk of testing individuals and classify them.

In Table 4, all the prediction accuracy of risk operators maintaining order are better than that with no order. For example, all R^* are better than R_HC.

7 Conclusions

Based on a decade research on risk index system of outsourced software project, 14 important attributes are extracted as the basis for modeling the risk operator. According to their mathematical properties, we propose a general risk function derived from IVIFS and software engineering, and divide them into 4 categories: incentive function, punitive function, incentive-punitive function and equilibrium function. This paper focuses on two kinds of incentive-punitive risk functions, and applies them to risk assessment of outsourced software project. In theory, we prove that if the risk function is a monotonically increasing function, then it will satisfy all the conditions as a risk function. By comparing the presented risk function with some conventional operators including R_M, R_NM, R_CT and R_HC, we conclude that it is easy to construct a reasonable and appropriate risk assessment model according to new risk functions by a combination of vectors maintaining order relations. And the experimental results illustrate the effectiveness of the presented risk function assessment model. However, Since the new risk function is suitable for special cases, the overall recognition effect is slightly inferior to the equilibrium risk function for the specific condition is not satisfied. The determination of weights is also an interesting topic. Moreover, the model can be extended to hesitant fuzzy set as [7].

Footnotes

Acknowledgments

This paper is funded by the National Natural Science Foundation of China (No. 71271061), Philosophy & Social Science Project (No. GD12XGL14) & Soft Science Project (No. 2015A070704051) & Natural Science Projects (No. 2014A030313575, 2016A030313688) of Guangdong Province, Science & Technology Fund of Guangdong Education Department (No. 2013KJCX0072), Philosophy & Social Science Project of Guangzhou (No. 14G41), Special Innovative Project (No. 15T21) & Key Team (No. TD1605) & Major Education Foundation (No. GYJYZDA14002) & Higher Education Foundation (No. 2016GDJYYJZD004) of Guangdong University of Foreign Studies, Climbing Plan Foundations of Guangdong (No. pdjh2015a0180, pdjh2016a0166).

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Incentive-punitive risk function with interval valued intuitionistic fuzzy information for outsourced software project risk assessment

Abstract

Keywords

1 Introduction

2 Conventional weighted order operators

Table 4 Average accuracy of IVIFS Risk Prediction Ranking Prediction Operator Accuracy Function Accuracy R1* 0.831 R M 0.804 R0* 0.827 R NM 0.796 R0.5* 0.819 R CT 0.831 R2* 0.823 R HC 0.723 R3* 0.807

Footnotes

Acknowledgments

References

Table 4
Average accuracy of IVIFS

Risk Prediction Ranking Prediction

Operator Accuracy Function Accuracy

R₁^* 0.831 R_M 0.804

R₀^* 0.827 R_NM 0.796

R_0.5^* 0.819 R_CT 0.831

R₂^* 0.823 R_HC 0.723

R₃^* 0.807