Multi-criteria software quality model selection based on divergence measure and score function

Abstract

Due to advancement of modern computing technology, software and its products have played an important role in the success and failure of any business. Nowadays, numerous researchers have focused their work on the development of new software quality models and its scopes. Evaluating software quality models based on several attributes is an essential and complex multi-criteria decision making problem (MCDM). To tackle this issue, several studies have been presented for the evaluation of software quality models by utilizing decision making and the idea of fuzzy sets (FSs). Present research develops an intuitionistic fuzzy (IF) improved score function method to deal with MCDM problems under intuitionistic fuzzy sets (IFSs). Further, the efficiency and feasibility of the proposed method is illustrated to choose the best software quality model under IF environment. To authorize the results, comparative study with previous developed approaches is discussed.

Keywords

Intuitionistic fuzzy sets divergence measure MCDM score function software quality model

1 Introduction

Over the last few decades, software has been proven to a key component for business systems, products and services. As a consequence, quality of software product is absolutely necessary for the success of any business. ISO9126 delineates software quality as “a set of characteristics of software quality by which its quality is portrayed and estimated”. For various problematic applications, software quality is required to run software smoothly. The process of assessing software quality recommends lots of benefits during the development process as it helps the developers to deliver high skilled software product. The software quality models are used to represent a more fixed and quantitative quality structure. Various quality models such as ISO9126, McCall, Boehm, Dromey, FURPS etc. have been developed to describe and manage the software quality [1 –5]. The quality models consist of number of quality parameters which reflects the quality of software products from view of that parameter, therefore, the quality models are considered as a MCDM model.

MCDM, a part of decision making, are widely appeared in business, politics and in our daily life problems. There are various situations and circumstances arise in our workplace that poses a MCDM problem. The purpose of MCDM is to choose an optimal option among a fixed set of options evaluated on several criteria by the consideration of decision experts. Several MCDM methods are proposed to evaluate the decision making problems in which the assessment information provided in crisp numbers. Due to time limitations and increasing intricacy of socio-economic atmosphere, the decision experts (DEs) are incapable to present information in crisp numbers form, therefore, the classical approaches are failed to handle the uncertain MCDM problems. FS theory, pioneered by Zadeh [6], has attracted the concentration of researchers and practitioners in many areas because the FSs are more capable to handle the uncertain information and knowledge. Atanassov [7] developed IFSs, which are distinguished by the belongingness and non-belongingness degrees. Various authors have concentrated their study on IFSs and its extensions to deal with ambiguity and vagueness of real-life issues [8 –18]. For instance, Mishra [19] developed trigonometric entropy and similarity measures based method to evaluate the township development problem with IFSs. Mishra et al. [11] originated an exponential information measure based technique to rank the service quality of vehicle insurance companies. Mishra & Rani [12] developed Shapley weighted divergence measures based VIKOR approach for correlative MCDM problems and applied to select an optimal cloud service provider with intuitionistic fuzzy information. Mishra et al. [13] originated a Jensen-exponential divergence measures based MCDM technique to deal with energy selection problem under intuitionistic fuzzy atmosphere. Rani et al. [16] proposed Pythagorean Fuzzy (PF) entropy and divergence measures and further, utilized to find criteria’s weights in Pythagorean fuzzy WASPAS approach. Next, Rani et al. [17] defined a novel PF-entropy measure and applied for the determination of criteria’s weights in VIKOR technique. Rani et al. [18] developed a PF-similarity measure based TOPSIS method to solve the real-life decision making problem.

Recently, several MCDM approaches viz. COPRAS [19, 20], ELECTRE [21], MABAC [22], PROMETHEE [23], TODIM [24], TOPSIS [25, 26], VIKOR [27, 28], WASPAS [29, 30], ARAS [31] etc have been presented with various fuzzy doctrines. For instance, Yu et al. [32] developed a new method to deal with multi-criteria group decision making (MCGDM) problems with unbalanced HFLTSs by considering the psychological behavior of decision makers. Zhang et al. [33] proposed an approach to deriving a priority weight vector from an incomplete hesitant fuzzy preference relation (HFPR) using the logarithmic least squares method. Based on the priority weight vector, the consistency index of an incomplete HFPR is defined, which calculates the average deviation between the priority weight vector and all elements of the incomplete HFPR. Zhang and Guo [34] focused on group decision making with intuitionistic multiplicative preference relation (IMPR) and analyzed the flaws of the consistency definition of an IMPR in previous work and then proposed a new definition to overcome the flaws. Mishra et al. [35] developed the IF-WASPAS method to compare the performance of telephone service providers in Madhya Pradesh (India). Ansari et al. [36] developed new information measures to detect the edge detection problem. Mishra et al. [30] proposed the hesitant fuzzy WASPAS method based on information measures to evaluate the green supplier selection problems.

Recently, Zhang and Dong [37] proposed a consensus-based group decision-making framework for failure mode and effect analysis (FMEA) with the aim of classifying FMs into several ordinal risk classes in which we assumed that FMEA participants provide their preferences in a linguistic way using possibilistic hesitant fuzzy linguistic information. Zhang et al. [38] defined the best additive consistency index, the worst additive consistency index and the average additive consistency index are defined to measure the consistency level of a HFPR. Zhang et al. [39] proposed a novel consensus framework based on Social Network Analysis (SNA) to deal with non-cooperative behaviors. In the proposed SNA-based consensus framework, a trust propagation and aggregation mechanism to yield experts’ weights from the social trust network is presented, and the obtained weights of experts are then integrated into the consensus-based MAGDM framework. Zhang et al. [40] filled the gap in the research literature on Consensus reaching processes (CRPs). To achieve this goal, firstly, a comprehensive review regarding the different approaches to CRP is reported, and a series of CRPs as the comparison objects are presented. Secondly, the following comparison criteria for measuring the efficiency of CPRs are proposed: the number of adjusted decision-makers, the number of adjusted alternatives, the number of adjusted preference values, the distance between the original and the adjusted preference information (adjustment cost) and the number of negotiation rounds required to reach consensus. Zhang et al. [41] developed a linguistic distribution-based optimization approach for converting comparative linguistic expressions (CLEs) into linguistic distribution assessments (LDAs), in which we assume that decision makers provide their opinions using preference relations with CLEs. Particularly, the proposed optimization approach is based on the use of a consistency-driven methodology, which seeks to minimize the inconsistency level of LDA preference relations obtained by transforming the original CLE preference relations elicited from decision makers.

Xu et al. [42] presented an improved definition of HFPRs, in which the values are not ordered for the hesitant fuzzy element and further, developed an additive consistency based estimation measure to normalize the HFPRs, with the help of which, a consensus model is proposed. Xu et al. [43] proposed a two-stage method to support the consensus reaching process for large-scale multi-attribute group decision making problems. Liu et al. [44] proposed a consensus model which considers overconfidence behaviors, and the paper mainly focuses on large scale group decision making (LSGDM) based on fuzzy preference relations with self-confidence. Liu et al. [45] firstly proposed a reliability index-based consensus reaching process for improve the efficiency of HF-LSGDM and secondly, developed an unreliable DM management method to be used in the reliability index-based consensus reaching process, based on the computation of DM’s opinion reliability index.

The proposed method is inspired by Weighted Distance Based Approximation (WDBA) [46], which measures the distance from the optimal point (best value of alternatives) and non-optimal point (worst value of alternatives). Finally, the alternatives are ranked by their suitability index (SI). The alternative having the least value of SI is ranked at first position and with the maximum value at last. In this method, the most favorable situation is represented by the ideal point (i.e. optimum point) and the least favorable situation is represented by the anti-ideal point (i.e. non-optimum point). For practical purposes, the ideal and anti-ideal points are defined as the best and the worst values, which exist within the range of values of attributes, respectively. The ideal point, then, is simply the alternative that has all the best values of attributes and the anti-ideal point is simply the alternative that has all the worst values of attributes. It may happen that a certain alternative has the best values for all attributes or the worst values for all attributes. Therefore, in this work, the ideal and anti-ideal points are also considered as feasible solution and they are used as reference in which other alternatives are quantitatively compared. The relative numerical difference resulting from comparison represents the effectiveness of alternatives known as the index score of the alternatives. The smaller index score, the closer the alternative resembles the optimal state, and vice versa. The WDBA approach has been applied in many fields such as E-learning websites selection [47], COTS component selection [48], and selection of software effort estimation [49], emergency decision making problems within interval-valued PFSs.

As the IFSs have capability to deal with imprecision and uncertainty of real-life problems, therefore, the present paper develops an improved score function (ISF) based MCDM method under intuitionistic fuzzy environment. In this method, the evaluation value of each alternative is given in IFNs form and the weight about the criteria is partially unknown. In order to compare the IFNs, some score and accuracy functions have been developed and utilized to rank the alternatives in MCDM problems. Further, the effectiveness and reliability of the developed method is demonstrated through a selection of a best Software quality model. Comparative study reveals the validity of the proposed approach. Therefore, the aim of this paper is to tackle the three challenges mentioned above by developing three MCDM approaches to managing evaluation information for IFSs, which has not only a great power in distinguishing the optimal alternative, but also can obtain an optimal alternative out of counter intuitive phenomena and parameter selection problems. According to the above discussions, the present work has following objectives and directions:

In this paper, improved score function based WDBA method is developed, which measures the overall performance of an alternative by divergences from the ideal point.

In order to compute the information measures (divergence measure, similarity measure and entropy) of two IFSs, we propose a new method for constructing of divergence measure. Later, we apply the divergence measure for decision making.

The remaining manuscript is summarized as follows: Section 2 scrutinizes fundamental ideas related to IFSs and score function. Section 3 presents an intuitionistic fuzzy improved score function method for MCDM problems. Section 4 illustrates the example of Software quality models selection problem, which proves the strength and usefulness of the developed approach. Section 5 summarizes the conclusion of the manuscript.

2 Preliminaries

In FS theory, the belongingness of an element is represented by a value which belongs to interval [0, 1], while non belongingness is essentially its complement. Although, in consequence, this assumption does not meet with human intuition. Accordingly, Atanassov [7] pioneered the idea of IFSs by portraying a belongingness and a non- belongingness functions.

Definition 2.1 [7]. An IFS X on V ={ u₁, u₂, . . . , u_n } is given by

$X = {〈 u_{i}, μ_{X} (u_{i}), ν_{X} (u_{i}) 〉 : u_{i} \in V},$ (1) where μ_X : V → [0, 1] and ν_X : V → [0, 1] show the belongingness degree and non-belongingness degree of u_i to X in V, with the condition

$\begin{matrix} 0 & ⩽ μ_{X} (u_{i}) ⩽ 1, 0 ⩽ ν_{X} (u_{i}) ⩽ 1 and \\ 0 & ⩽ μ_{X} (u_{i}) + ν_{X} (u_{i}) ⩽ 1, \forall u_{i} \in V \end{matrix}$ (2)

The intuitionistic index of an element u_i ∈ V to X is defined by $\begin{matrix} π_{X} (u_{i}) & = 1 - μ_{X} (u_{i}) - ν_{X} (u_{i}) and \\ 0 & \leq π_{X} (u_{i}) \leq 1, \forall u_{i} \in V . \end{matrix}$

For simplicity, Xu [51] characterized the intuitionistic fuzzy number (IFN) ξ = (μ_ξ, ν_ξ) which holds μ_ξ, ν_ξ ∈ [0, 1] and 0 ≤ μ_ξ + ν_ξ ≤ 1 .

Definition 2.2 [51]. Consider ξ_j = (μ_j, ν_j), j = 1, 2, . . . , n, be an IFN. Therefore,

$S (ξ_{j}) = (μ_{j} - ν_{j}), ℏ (ξ_{j}) = (μ_{j} + ν_{j}),$ (3) are the score and accuracy functions of the IFN ξ_j, respectively. At this point, $S (ξ_{j}) \in [- 1, 1]$ and ℏ (ξ_j) ∈ [0, 1] are called score degree and accuracy degree.

Since $S (ξ_{j}) \in [- 1, 1]$ , as numerous score values are combined through linear weighted summation and may be appeared that the positive score values are neutralize by the negative score values. Hence, Xu et al. [52] developed an improved score function for IFNs as follows:

Definition 2.3 [52, 53]. Let ξ_j = (μ_j, ν_j) be an IFN. Then

$\begin{matrix} S^{*} (ξ_{j}) = & \frac{1}{2} ((μ_{j} - ν_{j}) (2 - μ_{j} - ν_{j}) + 1), \\ ℏ^{\circ} (ξ_{j}) = 1 - ℏ (ξ_{j}), \end{matrix}$ (4) are normalized score and uncertainty functions, respectively. Evidently, $S^{*} (ξ_{j}) \in [0, 1]$ and ℏ° (ξ_j) ∈ [0, 1].

Suppose ξ₁ = (μ₁, ν₁) and ξ₂ = (μ₂, ν₂) are two IFNs. A system can be achieved effortlessly to compare any two IFNs with the use of normalized score $S^{*} (ξ_{j})$ and uncertainty functions ℏ° (ξ_j) , which as

If $S^{*} (ξ_{1}) > S^{*} (ξ_{2})$ , then ξ₁ > ξ₂,

If $S^{*} (ξ_{1}) = S^{*} (ξ_{2})$ , then

if ℏ° (ξ₁) > ℏ ° (ξ₂), then ξ₁ < ξ₂;

if ℏ° (ξ₁) = ℏ ° (ξ₂) , then ξ₁ = ξ₂.

Definition 2.4 [51]: Suppose ξ_j = (μ_j, ν_j), j = 1, 2, . . . , n be IFNs, then IFWAO is given by

$\begin{matrix} {IFWA}_{w} (ξ_{1}, ξ_{2}, . . ., ξ_{n}) \\ = [1 - \prod_{j = 1}^{n} {(1 - μ_{j})}^{w_{j}}, \prod_{j = 1}^{n} ν_{j}^{w_{j}}], \end{matrix}$ (5) where w_j = (w₁, w₂, . . . , w_n) ^T is a weight vector of ξ_j, j = 1 (1) n, with $\sum_{j = 1}^{n} w_{j} = 1$ , w_j ∈ [0, 1] .

Divergence measure is a well-known mode to compute the degree of inequity. Initially, Vlachos & Sergiadis [54] pioneered the idea of IF-divergence measure. In a while, Montes et al. [55] presented a novel explanation of IF-divergence measure, which as

Definition 2.5 [55] A divergence measure $L : IFSs (V) \times IFSs (V) \to ℝ$ is a mapping which holds the following:

L (G, H) = L (H, G) ;

L (G, H) = 0 iff G = H ;

L (G ∩ I, H ∩ I) ≤ L (G, H) for every I∈ IFS (V) ;

L (G ∪ I, H ∪ I) ≤ L (G, H) for every I ∈ IFS (V).

Definition 2.6 [13]. Consider that G, H ∈ IFSs (V),

$\begin{matrix} L (G, H) \\ = \frac{1}{2 n (1 - e^{- 1})} \sum_{i = 1}^{n} [{e^{- (\frac{(μ_{H} (u_{i}) - μ_{G} (u_{i})) + (ν_{G} (u_{i}) - ν_{H} (u_{i}))}{2})} + e^{- (\frac{(μ_{G} (u_{i}) - μ_{H} v) + (ν_{H} (u_{i}) - ν_{G} (u_{i}))}{2})} - 2} I_{[μ_{G} (u_{i}) ⩾ ν_{G} (u_{i})]} \\ + {e^{- (\frac{(μ_{G} (u_{i}) - μ_{H} (u_{i})) + (ν_{H} (u_{i}) - ν_{G} (u_{i}))}{2})} + e^{- (\frac{(μ_{H} (u_{i}) - μ_{G} (u_{i})) + (ν_{G} (u_{i}) - ν_{H} (u_{i}))}{2})} - 2} I_{[μ_{G} (u_{i}) < ν_{G} (u_{i})]}] . \end{matrix}$ (6)

3 Intuitionistic Fuzzy Improved Score Function (IF-ISF) method for MCDM

In the process of MCDM, the main purpose is to select the best option from a set S ={ S₁, S₂, . . . , S_m } of the options assessed on criterion set C ={ C₁, C₂, . . . , C_n }. Consider a set of ℓ decision experts (DEs) D ={ D₁, D₂, . . . , D_ℓ } to find an optimal option(s). The procedure of the novel IF-ISF method is elaborated as below:

Step 1: Establish the IF decision matrix (Z_k) by regarding a group of decision experts (DEs).

Construct performance evaluation matrix $Z_{k} = {[{\bar{z}}_{ij}^{(k)}]}_{m \times n}$ for each DE versus criterion set.

$\begin{matrix} \begin{matrix} C_{1} & \dots & C_{n} \end{matrix} \\ Z_{k} = {({\bar{z}}_{ij}^{k})}_{m \times n} = \begin{matrix} S_{1} \\ ⋮ \\ S_{m} \end{matrix} {[\begin{matrix} 〈 μ_{11}^{k}, ν_{11}^{k} 〉 & \dots & 〈 μ_{1 n}^{k}, ν_{1 n}^{k} 〉 \\ ⋮ & ⋱ & ⋮ \\ 〈 μ_{m 1}^{k}, ν_{m 1}^{k} 〉 & \dots & 〈 μ_{mn}^{k}, ν_{mn}^{k} 〉 \end{matrix}]}_{m \times n}, \\ \forall k . \end{matrix}$ (7)

Step 2: Determine crisp DEs weights (ϖ_k)

DEs are provided importance ratings by the experts based on their expertise as G_k = (μ_k, ν_k, π_k) k = 1 (1)ℓ. To reflect their relative importance in the decision making process, numeric DEs weights are obtained as follows [56]:

$\begin{matrix} ϖ_{k} = & \frac{μ_{k} (2 - μ_{k} - ν_{k})}{\sum_{k = 1}^{ℓ} (μ_{k} (2 - μ_{k} - ν_{k}))}, \\ where \sum_{k = 1}^{ℓ} ϖ_{k} = 1 . \end{matrix}$ (8)

Step 3: Obtain the aggregated intuitionistic fuzzy (AIF) decision matrix

Suppose $Z = (z_{ij}^{k})$ is the decision matrix of k^th DE into the form of decision linguistic variables and then transformed to the IFN such that ${\bar{z}}_{ij}^{k} = 〈 μ_{ij}^{k}, ν_{ij}^{k} 〉$ , k = 1, 2, . . . , ℓ. Next, to combine each of the individual decision into one group decision, construct AIF-decision matrix.

Consider $ℤ = {[z_{ij}]}_{m \times n}$ , wherein z_ij =〈 μ_ij, ν_ij 〉, i = 1 (1) m, j = 1, 2, . . . , n, be the AIF-decision matrix, where $ℤ = \oplus_{k = 1}^{ℓ} ϖ_{k} {\bar{z}}_{ij}^{k}$ and $z_{ij} = 〈 1 - \prod_{k = 1}^{ℓ} {(1 - μ_{ij}^{k})}^{ϖ_{k}}, \prod_{k = 1}^{ℓ} {(ν_{ij}^{k})}^{ϖ_{k}} 〉 .$ (9)

Step 4: Create the normalized AIF-decision matrix

Normalize the AIF- decision matrix $ℤ = {[z_{ij}]}_{m \times n}$ and expressed by $ℕ = {[ς_{ij}]}_{m \times n}$ , where

$\begin{matrix} ς_{ij} & = 〈 {\tilde{μ}}_{ij}, {\tilde{ν}}_{ij} 〉 \\ = {\begin{matrix} z_{ij} = 〈 μ_{ij}, ν_{ij} 〉, for benefit criterion \\ {(z_{ij})}^{c} = 〈 ν_{ij}, μ_{ij} 〉, for cost criterion \end{matrix} . \end{matrix}$ (10)

Step 5: On the basis of ISF, transfer this matrix into the score matrix $S^{*}$ which as

$\begin{matrix} \begin{matrix} C_{1} & C_{2} & \dots & C_{n} \end{matrix} \\ S^{*} = S^{*} (S_{ij}) = \begin{matrix} S_{1} \\ S_{2} \\ ⋮ \\ S_{m} \end{matrix} [\begin{matrix} S^{*} (ς_{11}) & S^{*} (ς_{12}) & \dots & S^{*} (ς_{1 n}) \\ S^{*} (ς_{21}) & S^{*} (ς_{22}) & ⋮ & S^{*} (ς_{2 n}) \\ ⋮ & ⋮ & ⋱ & \dots \\ S^{*} (ς_{m 1}) & S^{*} (ς_{m 2}) & \dots & S^{*} (ς_{mn}) \end{matrix}], \end{matrix}$ (11)

where $S^{*} (S_{ij}) (i = 1, 2, . . ., m; j = 1, 2, . . ., n,)$ are computed by using Definition 2.3.

Since it has been evident that the final ranking result of alternatives highly depends on the criterion weights and hence the proper assessment of the criterion weights plays a dominant role in making decisions. Thus, in the MCDM process, a weighted sum of each option is evaluated by multiplying the score function for each criterion and its assigned weight and called as suitability function $ℚ (S_{i})$ , defined as $ℚ (S_{i}) = \sum_{j = 1}^{n} w_{j} S^{*} (S_{ij}); i = 1, 2, . . ., m .$ (12)

Here, w = (w₁, w₂, . . . , w_n) ^T denotes the normalized weight vector of the criterion C_j (j = 1, 2, . . . , n) and a subset of the weight information is denoted by $ℂ$ . The suitability function $ℚ (S_{i})$ has been utilized to identify the degree to which an option fulfils the DEs conditions. Also, a sensible weight vector should create the overall value of suitability functions of all options as large as possible. This principal leads to the formulation of the mathematical programming model for determining the weight vector as follows:

$(M - I) : {\begin{matrix} max ϑ = \sum_{i = 1}^{m} ℚ (S_{i}) = \sum_{i = 1}^{m} \sum_{j = 1}^{n} w_{j} S^{*} (S_{ij}) \\ s . t . \sum_{j = 1}^{n} w_{j} = 1, w_{j} \geq 0 and w_{j} \in ℂ . \end{matrix}$ (13)

Here, $ℚ (S_{i})$ represents the overall score function for each option S_i (i = 1, 2, . . . , m). After solving model (M-I), the weight vector w = (w₁, w₂, . . . , w_n) ^T is obtained.

Step 6: Compute the performance degree of alternative

The ranking method [57] for ranking IFNs γ_i = ξ_i (w) = (μ_i, ν_i) , i = 1, 2, . . . , m is implemented, where the overall criterion value of each option S_i is

$\begin{matrix} ξ_{i} (w) & = \sum_{j = 1}^{n} w_{j} ς_{ij} \\ = (\sum_{j = 1}^{n} w_{j} {\tilde{μ}}_{ij}, \sum_{j = 1}^{n} w_{j} {\tilde{ν}}_{ij}, \sum_{j = 1}^{n} w_{j} {\tilde{π}}_{ij}), \\ i = 1, 2, . . ., m . \end{matrix}$ (14)

Define the performance degree

$φ (γ_{i}) = 0.5 (1 + π_{i}) L (γ^{*}, γ_{i}), i = 1, 2, . . ., m,$ (15) where L (γ^*, γ_i) , i = 1 (1) m, is divergence measure for IFNs given by (6).

$γ^{*} = {(μ_{1 *}, ν_{1 *}), (μ_{1 *}, ν_{1 *}), . . ., (μ_{m *}, ν_{m *})},$ (16)

Such that

$\begin{matrix} (μ_{i *}, ν_{i *}) = & {\begin{matrix} (max_{j} μ_{i}, min_{j} ν_{i}), if C_{j} \in C_{b} \\ (min_{j} μ_{i}, max_{j} ν_{i}), if C_{j} \in C_{n} \end{matrix}, \\ i = 1 (1) m . \end{matrix}$ (17)

The smaller value of φ (γ_i) depicts the better the overall interval valued intuitionistic fuzzy preference value γ_i [57].

Step 7: Choose the optimal alternative(s) based on the performance degree φ (γ_i) .

Step 8: End

4 Software quality model selection problem for MCDM

In this section, a case study is presented for selecting the best Software quality model among a set of alternative quality models, which shows the feasibility of the proposed IF-ISF method.

As an initial step, a set of three decision experts (DEs) (D_i : i = 1, 2, 3) is formed for the process of Software quality model selection problem under intuitionistic fuzzy environment. After preliminary screening, the DEs determine three Software quality models, which are Boehm model (S₁), McCall Model (S₂) and ISO9126 (S₃). The preferences of each alternative are given by the DEs with respect to set of three criteria that are product operation, product revision and product transition factors. These criteria are divided into four sub-criteria, which are trustworthiness (C₁), effectiveness (C₂), maintainability (C₃) and portability (C₄). The descriptions of the identified sub-criteria are given as follows:

Trustworthiness: Trustworthiness pacts with service breakdown. They decide the utmost permitted breakdown rate of the software system, and can refer to the whole system or to one or more of its parts.

Effectiveness: Effectiveness handles the hardware assets required to execute dissimilar functions of the software system. It also pacts with the time between recharging of the system’s portable units.

Maintainability: Maintainability considers the efforts that will be required by clients and maintenance employees for the identification of the causes of software breakdown, for the correction of breakdowns and for the verification of the improvements.

Portability: This decisive factor tends to the variation of a software system to another atmosphere that consists of diverse hardware, dissimilar operating systems, and so forth. The software should be probable to persist with the use of same basic software in different circumstances.

Now, the proposed method is utilized to solve the above mentioned problem, given as below:

Step 1: Suppose that these Software quality models (S_i : i = 1, 2, 3) are assessed by a set of DEs with respect to each criterion (C_i : i = 1, 2, 3, 4) and the assessment of each models are given in the form IFNs (see Table 1).

Table 1
Assessment $Z_{k} = {[{\bar{z}}_{ij}^{(k)}]}_{3 \times 4}$ given by DEs D_k : k = 1, 2, 3

D ₁ C ₁ C ₂ C ₃ C ₄

S ₁ (0.60, 0.25, 0.15) (0.55, 0.35, 0.05) (0.62, 0.34, 0.04) (0.70, 0.22, 0.08)

S ₂ (0.68, 0.20, 0.12) (0.63, 0.30, 0.07) (0.55, 0.38, 0.07) (0.59, 0.35, 0.06)

S ₃ (0.64, 0.25, 0.11) (0.71, 0.18, 0.11) (0.58, 0.28, 0.14) (0.62, 0.25, 0.13)

D ₂

S ₁ (0.63, 0.15, 0.22) (0.68, 0.19, 0.14) (0.72, 0.15, 0.13) (0.69, 0.20, 0.11)

S ₂ (0.74, 0.16, 0.10) (0.52, 0.28, 0.20) (0.57, 0.26,0.17) (0.63, 0.28, 0.09)

S ₃ (0.76, 0.15, 0.09) (0.61, 0.19, 0.20) (0.56, 0.27, 0.17) (0.77, 0.13, 0.10)

D ₃

S ₁ (0.61, 0.23, 0.16) (0.84, 0.11, 0.05) (0.54, 0.28, 0.18) (0.59, 0.26, 0.15)

S ₂ (0.67, 0.22, 0.11) (0.71, 0.17, 0.12) (0.74, 0.20, 0.06) (0.61,0.16, 0.23)

S ₃ (0.66, 0.24, 0.10) (0.74, 0.17, 0.09) (0.64, 0.19, 0.17) (0.69, 0.18,0.13)

D ₁	C ₁	C ₂	C ₃	C ₄
S ₁	(0.60, 0.25, 0.15)	(0.55, 0.35, 0.05)	(0.62, 0.34, 0.04)	(0.70, 0.22, 0.08)
S ₂	(0.68, 0.20, 0.12)	(0.63, 0.30, 0.07)	(0.55, 0.38, 0.07)	(0.59, 0.35, 0.06)
S ₃	(0.64, 0.25, 0.11)	(0.71, 0.18, 0.11)	(0.58, 0.28, 0.14)	(0.62, 0.25, 0.13)
D ₂
S ₁	(0.63, 0.15, 0.22)	(0.68, 0.19, 0.14)	(0.72, 0.15, 0.13)	(0.69, 0.20, 0.11)
S ₂	(0.74, 0.16, 0.10)	(0.52, 0.28, 0.20)	(0.57, 0.26,0.17)	(0.63, 0.28, 0.09)
S ₃	(0.76, 0.15, 0.09)	(0.61, 0.19, 0.20)	(0.56, 0.27, 0.17)	(0.77, 0.13, 0.10)
D ₃
S ₁	(0.61, 0.23, 0.16)	(0.84, 0.11, 0.05)	(0.54, 0.28, 0.18)	(0.59, 0.26, 0.15)
S ₂	(0.67, 0.22, 0.11)	(0.71, 0.17, 0.12)	(0.74, 0.20, 0.06)	(0.61,0.16, 0.23)
S ₃	(0.66, 0.24, 0.10)	(0.74, 0.17, 0.09)	(0.64, 0.19, 0.17)	(0.69, 0.18,0.13)

Step 2: Furthermore, suppose that the weights of DEs are IFNs and are expressed as [0.32, 0.52], [0.25, 0.55], [0.28, 0.50].

Since DEs weights are given in the form of IFNs, therefore, the crisp DMs weights ϖ_k : k = 1, 2, 3 have been obtained using Equation (8), which as

$ϖ_{k} = {0 . 3665, 0 . 2962, 0 . 3373} .$ (18)

Step 3: Using Equation (9), the AIF-decision matrix is computed. Table 2 presents the result.

Table 2

Aggregation of individual assessment $ℤ = {[z_{ij}]}_{3 \times 4}$ over DEs weights ϖ_k : k = 1, 2, 3

	C ₁	C ₂	C ₃	C ₄
S ₁	(0.6125, 0.2089)	(0.7130, 0.1977)	(0.6298, 0.2499)	(0.6634, 0.2263)
S ₂	(0.6959, 0.1933)	(0.6319, 0.2427)	(0.6310, 0.2735)	(0.6089, 0.2516)
S ₃	(0.6868, 0.2120)	(0.6949, 0.1794)	(0.5957, 0.2430)	(0.6942, 0.1844)

Step 4: Since all the criteria (C_i : i = 1, 2, 3, 4) are of benefit types, therefore, no need to normalize the AIF-decision matrix given in Table 2.

Step 5: By utilizing an ISF given in Eq. (4), convert the above matrix into their collective score matrix as

$\begin{matrix} \begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} \end{matrix} \\ S^{*} = S^{*} (S_{ij}) = \begin{matrix} S_{1} \\ S_{2} \\ S_{3} \end{matrix} [\begin{matrix} 0.7378 & 0.7807 & 0.7128 & 0.7427 \\ 0.7791 & 0.7190 & 0.6958 & 0.7036 \\ 0.7614 & 0.7901 & 0.7048 & 0.7858 \end{matrix}] \end{matrix}$ (19)

Step 6: Suppose the information about the criteria weights is partially known and is given

$\begin{matrix} ℂ = & {w = {(w_{1}, w_{2}, w_{3}, w_{4})}^{T} : 0.2 \leq w_{1} \leq 0.3, \\ 0.23 \leq w_{2} \leq 0.32, 0.18 \leq w_{3} \leq 0.25, \\ 0.25 \leq w_{4} \leq 0.33, \sum_{j =}^{4} w_{j} = 1 and w_{j} \geq 0} . \end{matrix}$ (20)

On the basis of this information, an optimization model is originated as

$\begin{matrix} max ϑ = 2.2783 w_{1} + 2.2898 w_{2} + 2.1134 w_{3} \\ + 2.2321 w_{4} \\ s . t . 0.2 \leq w_{1} \leq 0 . 3, 0.23 \leq w_{2} \leq 0.32, \\ 0.18 \leq w_{3} \leq 0 . 25, 0 . 25 \leq w_{4} \leq 0.33, \\ \sum_{j = 1}^{4} w_{j} = 1, and w_{1}, w_{2}, w_{3}, w_{4} \geq 0 \end{matrix}$ (21) and thus, the computed criteria’s weight vector is w = (0.25, 0.32, 0.18, 0.25) ^T.

The overall criterion value ξ_i (w) of the alternative S_i (i = 1 (1) 3) is calculated by (14) and is given by

$\begin{matrix} ξ_{1} (w) & = (0 . 6630, 0 . 2163), ξ_{2} (w) \\ = (0 . 6435, 0 . 2364), ξ_{3} (w) = (0 . 6767, 0 . 1989) . \end{matrix}$

Step 7: Next, based on IF-divergence measure, we apply the ranking method to evaluate the performance degree of alternative by using (15) as follows: $\begin{matrix} φ (ξ_{1} (w)) & = 0.0005, φ (ξ_{2} (w)) = 0.0010, \\ φ (ξ_{3} (w)) & = 0.0002, \end{matrix}$ where γ^* ={ (0 . 7130, 0 . 1977) , (0 . 6959, 0 . 1933) , (0 . 6949, 0 . 1794) } is the largest IFN (or PIV), is calculated by (16) and (17).

Furthermore, the three alternatives S_i (i = 1 (1) 3) are rank in accordance with ξ_i (w) , (i = 1 (1) 3) and is given by S₃ ≻ S₁ ≻ S₂, where ≻ means “be superior to.” Hence S₃ is the most desirable alternative.

The ranking of these Software quality modes is compared with Kohli & Sehra [2] and Sehra et al. [4] methods. The comparison of ranking order by proposed and existing methods is presented in Table 3.

Table 3

Comparison of proposed method with previous developed approaches

Approaches	Ranking order	Benchmark	DEs weights	Criteria weight	Optimal options
Kohli &Sehra [2]	S₁ ≻ S₂ ≻ S₃	FSs	Not considered	Fuzzy AHP method	S ₁
Sehra et al. [4]	S₃ ≻ S₁ ≻ S₂	FSs	Not considered	Fuzzy AHP method	S ₃
Kumari et al [35] method	S₃ ≻ S₁ ≻ S₂	IFSs	Not considered	Divergence measure with Shapley-TOPSIS method	S ₃
Rani et al [36] method	S₃ ≻ S₁ ≻ S₂	PFSs	Computed	Similarity measure with TOPSIS method	S ₃
Proposed method	S₃ ≻ S₁ ≻ S₂	IFSs	Computed	Improved score function based linear programming model	S ₃

Comparing the mathematical results, we observe that the grading obtained from proposed method and Sehra et al. [4] method are similar but the Kohli & Sehra [2] produces quite different ranking.

The main benefits of the developed IF-ISF method are:

The DEs weights are computed, which was previously not considered.

The partially unknown weight information of criterion is determined by improved score function based linear programming model.

To deal with MCDM uncertainty problems, all the inputs, viz., the evaluations of alternative on criteria by several DEs, the weights of DEs, and the criteria weights are considered in terms of IFNs.

In the evaluation of Software quality models, IFSs are more capable than FSs. Thus, the proposed approach is more feasible and reliable than existing methods.

In the terms of generating the same preference rankings, proposed method is also important tool to decide in a short time as well as to make effective and require relatively less effort than other IF-based methods for taking decisions in MCDM environment.

The IF-ISF-WDBA method is to provide more accurate ranking results using the hesitancy degree is treated in parallel important in whole methodology and ranking of alternatives is obtained using transaction values of all three parameters.

5 Conclusions

In the selection of Software quality model, the most important point which should be remembers that the quality model must assure the requirements of Software products. This paper develops an approach named as IF-ISF, based on linear programming model for MCDM with partially weight information. This method has employed to find the best Software quality model under intuitionistic fuzzy doctrines. In this method, the improved weighted score function has utilized to rank the Software quality models and find the optimal one. Finally, the validity of proposed approach has been demonstrated through a comparison with some existing methods.

In the future, we will extend the IF-MABAC method by utilizing different benchmarks, namely, ANP, AHP, SWARA, and Choquet integral, in the MCDM procedure. Furthermore, the developed method can be used for solving the real-world problems of other application areas such as supplier selection, material selection, and digital supply chain problems to illustrate its strength and effectiveness.

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