Complex intuitionistic fuzzy soft SWARA - COPRAS approach: An application of ERP software selection

1.   Introduction

Uncertainties and vagueness are characteristics that are prevalent in problems which occurring in science, engineering, medical science, economics, decision making, etc. To exceed these, some of the theories were designated like fuzzy set (FS) ^[38], intuitionistic fuzzy set (IFS) ^[2], soft set (SS) ^[23] and intuitionistic fuzzy soft set (IFSS) ^[18]. MCDM problems often require the decision makers to give evaluation information about the alternative and the criteria with a FS, soft set, IFS and other extended sets ^{[12,15,20,36,37]}. Under these existing sets, there is a difficulty to handle the seasonality or periodicity that to be in many real life problems. The complex numbers can be used to model the periodicity of the elements by its phase term and also complex numbers can represent the two dimensional information. The complex valued models have been widely used in various fields of applications ^[3,34,35].

In this regard, Ramot et al. ^[28] introduced a novel concept known as complex fuzzy sets (CFSs), which included the amplitude and phase terms. By adding the degree of non-belongingness to CFSs, Alkouri et al. ^[1] proposed the idea of complex intuitionistic fuzzy sets (CIFSs). Further, Kumar et al. ^[13] defined the CIFSSs (complex intuitionistic fuzzy soft sets) concept. In recent times, CIFS is a powerful context means to represent the two dimensional data arisen in natural complicated problems ^[6,29]. Consequently, numerous methodologies and theories have been established under the CFSs extensions ^{[9,16,17,19,24,25,26,27,32]}. In this work, a novel MCDM technique is proposed to select the ideal choice in which the data is represented in the form of CIFSS environment.

Zavadskas et al. ^[39] defined a novel approach known as COPRAS (COmplex PRoportional ASsessment), which is a realistic decision making framework. The advantages of the COPRAS framework are: (ⅰ) it contains the ratios of benefit and cost solutions simultaneously; (ⅱ) it is a valuable and flexible method to solve the problems. Several works have extended the COPRAS method in various uncertain contexts. Recently, Garg ^[7] proposed the COPRAS method in possibility intuitionistic fuzzy soft sets environment for the decision problem. Also, Mishra et al. ^[14,22] introduced the COPRAS Method in various fuzzy extensions to solve the decision making. The weights of the criteria are key factors in the decision process. There are two kinds of criteria weights: (ⅰ) subjective; (ⅱ) objective. The objective weights are computed from the data given in decision matrices and the subjective weights are evaluated based on the data given by the experts. To evaluate the subjective criteria weights, Kersuliene et al. ^[11] introduced a novel approach called SWARA, which has a least computational complexity. The SWARA and fuzzy COPRAS approach was integrated by Ighravwe and Oke ^[4] to select the maintenance technician. Further, the combined SWARA and COPRAS approach is developed in various domain such as hesitant fuzzy sets ^[30] and intuitionistic fuzzy sets (IFSs) ^[21]. These approaches had their own significance in the evaluation of weight and preference ranking of an alternative, respectively.

The existing studies such as IFS, IFSS etc., have been widely applicable in many fields, although it has been restricted in nature. In such theories, the information collected related to the object is deal with only one dimension information at a time, which may result to loss of some information at a particular time. In everyday life, we come across complex phenomena where it becomes necessary to add the other dimension to the membership and non-membership grades. By defining this other dimension, the complete information can be projected in single set, and hence, information loss can be ignored. To illustrate the importance of the phase term, consider an automotive manufacturing company in India, where they decide to select the ERP software for improving their business activities. The company consults an expert team regarding (ⅰ) overall rating about an alternative; (ⅱ) corresponding latest version of ERP software. This is a two-dimensional problem and this is unable to done accurately using IFS and IFSS ^[2,18] structure. To the best of our knowledge, this work develops a new framework called Complex Intuitionistic Fuzzy Soft-SWARA-COPRAS (CIFS-SWARA-COPRAS) with CIFSS domain for solving the two dimensional problem. In CIFSS atmosphere, the amplitude term represents the expert's decision with respect to the alternative and the phase term represents the expert's decision with respect to the current version of ERP software.

An ERP system is the brain of the company's technology system. Because ERP combines all features of a business involving development of product, manufacturing, retail and sales. ERP software selection is one of the most important decision making issues covering both qualitative and quantitative criterion for organizations. The authors of ^[10], in their conjoint work of 1008 calculation completed in 126 organizations state that reliability, functionality, ease of use and ease of customization are considered as the most important criteria. Based on these criteria, an automotive manufacturing company in India, where they decides to select the ERP software by using CIFS-SWARA-COPRAS technique.

The main contributions of this paper include:

1) A new framework named as CIFS-SWARA-COPRAS is introduced in the work. In this framework, criteria weights are evaluated by the SWARA technique, and the ranking of alternatives is determined by the COPRAS method using CIFSs information.

2) In the proposed technique, the weights are calculated by using the SWARA method which include the exactness of experts' opinion and the ranking of an alternative is determined by using the COPRAS approach.

3) To illustrate the applicability of the presented framework, an empirical case study of ERP software selection problem is examined under CIFSSs environment. The problem is two dimensional, that is to find the ideal ERP software with its current version.

4) A sensitivity analysis and comparative study are presented to show the validity and stability of the defined methodology.

The rest of this paper is organized as follows: Section 2 describes the notion of complex fuzzy sets and complex intuitionistic fuzzy soft sets. Section 3 proposes the CIFS-SWARA-COPRAS method under the CIFSS-environment. The applicability of the stated method is demonstrated through a case study of ERP software selection in Section 4. Also, a comparative study and sensitivity analysis is conducted to strengthen the results over the existing studies results. At last, Section 5 concludes the work.

2.   Preliminaries

In this section, we review the notions of CFS, CIFS and CIFSS.

Definition 2.1. ^[28] A complex fuzzy set (CFS) $C$ over $U$ is distinguished by a membership function $\mu_{C}(u)$ that belongs to the unit circle in the complex plane and is denoted by $\mu_{C}(u) = r_{C}(u)e^{is_{C}(u)}$, where $r_{C}(u)$ and $s_{C}(u)$ are both real-valued, $r_{C}(u)\in [0, 1]$ and $i = \sqrt{-1}$. A CFS $C$ is represented by,

Definition 2.2. ^[13] Let $CIFS(U)$ denotes the set of all CIFs over $U$, $E$ be a set of criteria. Then

is a complex intuitionistic fuzzy soft set (CIFSS) over U, where $\widetilde{F}: E \longrightarrow CIFS(U)$.

For all $\epsilon \in E$,

where $\mu_{\widetilde{F}(\epsilon)}:U\longrightarrow\{a|a\in \mathbb{C}, |a|\leq 1\}$ and $\nu_{\widetilde{F}(\epsilon)}:U\longrightarrow\{a'|a'\in \mathbb{C}, |a'|\leq 1\}$, $|\mu_{\widetilde{F}(\epsilon)}(u)+ \nu_{\widetilde{F}(\epsilon)}(u)|\leq 1$.

For convince, we denote $\delta = \left(re^{is}, \tau e^{i\psi}\right)$ and called as CIFN ("complex intuitionistic fuzzy number") with the condition that $r, \tau, s, \psi\in [0, 1]$ and $\mid re^{is} + \tau e^{i\psi}\mid \le 1$.

Definition 2.3. ^[6] Let $\delta = \left(re^{is}, \tau e^{i\psi}\right)$ be a CIFN. Then the score function of $\delta$ is defined as

and the accuracy function is

It is clearly seen that $S(\delta)\in[-2, 2]$ and $H(\delta)\in[0, 2]$.

Definition 2.4. Let $\delta$ be a CIFN. Then the normalized score function of $\delta$ is defined as

where $S(\delta)$ is the score function of $\delta$ and $\mathbb{S}^{*}(\delta)\in[0, 1].$

Definition 2.5. ^[6] Let $\delta_{j} = \left(r_j e^{is_j}, \eta_j e^{i\psi_j}\right) (j = 1, 2, ..., n)$ be CIFNs. The aggregated value of these CIFNs is obtained by using the complex intuitionistic fuzzy weighted averaging operator (CIFWA) and given as

where $w = (w_{1}, w_{2}, ..., w_{n})^{T}$ is a weight vector of $\delta_{j}$ with $w_{j} > 0$ and $\sum\limits_{j = 1}^{n}w_{j} = 1$.

3.   Proposed CIFS-SWARA-COPRAS method

In this section, we propose the concept of CIFS-SWARA-COPRAS method and the working framework of it in the following steps:

Step 1: Originate the data about the alternatives according to the criteria.

Let $A = \{a_{1}, a_{2}, ..., a_{m}\}$ be a collection of decision criteria and $U = \{x_{1}, x_{2}, ..., x_{n}\}$ denotes the set of alternatives and let $DE = \{E_{1}, E_{2}, ..., E_{l}\}$ be a set of decision experts for the decision process and $W = \{w_{1}, w_{2}, ..., w_{l}\}$ represents the weights for the decision experts. Suppose $X$ be the initial CIFSs decision matrix

where $x_{ij}^{(k)} = \left(r_{ij}^{(k)} e^{is_{ij}^{(k)}}, \eta_{ij}^{(k)} e^{i\psi_{ij}^{(k)}}\right)$ denotes the initial value for $x_{i}$ as to the criteria $a_{j}$ given by the $k^{th}$ expert in terms of CIFS set, for $j = 1, 2, ..., m$, $i = 1, 2, ..., n$ and $k = 1, 2, ..., l$.

Step 2: Generate the aggregated CIFSS decision matrix.

Compute the aggregated value of the expert $x_{ij}^{(k)} = \left(r_{ij}^{(k)} e^{is_{ij}^{(k)}}, \eta_{ij}^{(k)} e^{i\psi_{ij}^{(k)}}\right)$ for $k = 1, 2, ..., l$ into CIFSS decision matrix $\widehat{X} = (\widehat{x}_{ij})_{n\times m}$ by utilizing the Definition 2.5. The obtained values of $\widehat{x}_{ij} = \left(r_{ij} e^{is_{ij}}, \eta_{ij} e^{i\psi_{ij}}\right)$ are as follows

Step 3: Evaluate the weights for the criteria by using SWARA method.

The weights of the criteria are computed by using SWARA method, whose steps are summarized as follows.

ⅰ) Evaluate the crisp values $w_{j}$ for the criteria according to the experts' preference and find the normalized score values of $w_{j}$ (i.e., $\mathbb{S}^{*}(w_{j})$), by using the Definition 2.4.

ⅱ) Order the criteria from the most to the least score value.

ⅲ) Compute the comparative value $k_{j}$ for the ordered score value, which starts from the second place by differencing the $j^{th}$ value and $(j-1)^{th}$ value.

ⅳ) Calculate the comparative coefficient $p_{j}$ by

ⅴ) Evaluate the recalculated weights $q_{j}$ by

ⅵ) Estimate the criteria weights $w_{j}$ by using the following equation

Step 4: Determine the optimisation indexes $\alpha_{i}^{+}$ and $\alpha_{i}^{-}$ for $A^{+}$ and $A^{-}$, respectively.

Let us assume $A^{+} = \{a_1, a_2, ..., a_r\}$ where $r < m$ be the set of positive (beneficial) criteria. Then we calculate the optimisation index ($\alpha_{i}^{+}$) for each alternative $x_{i}$, as follows

Let $A^{-} = \{a_{r+1}, a_{r+2}, ..., a_m\}$ be the non-beneficial (negative) criteria for the decision problem. Then we determine the optimization index ($\alpha_{i}^{-}$) for each alternative $x_{i}$, as

Step 5: Calculate the priority value ($\mathbb{Q}_{i}$), for each $x_{i}$.

By using COPRAS technique, we compute the priority value $\mathbb{Q}_{i}$, for every $x_{i}$ as:

Step 6: Make the endmost decision based on utility degree $\gamma_{i}$.

Calculate the utility degree $\gamma_{i}$, for every $x_{i}$ by

where $Q_{max}\neq 0$ is the largest value of $\mathbb{Q}_{i}$. Then, rank the utility degree in which the highest utility degree is ranked as first, while least utility degree is ranked as last. Finally, make the conclusion based on the utility degree $\gamma_{i}$.

The implementation framework of the CIFS-SWARA-COPRAS method is shown in Figure 1.

4.   A case study

To demonstrate the working of the proposed CIFS-SWARA-COPRAS method, we provide a numerical example related to ERP software selection, which can be read as follows.

4.1.   An empirical study: ERP software selection

The proposed CIFS-SWARA-COPRAS algorithm is applied to a Enterprise Resource Planning (ERP) software selection process for the automotive manufacturing company in India. In the manufacturing company, the operation managers need to buy an ERP software system for increasing the day-to-day business activities such as production management, accounting, manufacturing execution, procurement, supply chain operations and compliance. Initially, a decision expert team of 3 $DE = \{E_{1}, E_{2}, E_{3}\}$ consisting of a senior representative and a functional expert and a senior manager was constituted. Then, the expert team decides seven criteria (Functionality, Reliability, Vendor Viability, Cost, Support and Training, Ease of Use and Difficulty in Customization and Implementation) based on their experiences and studies in the literature. Accordingly, the criteria and its direction for the ERP selection process is given in Table 1. In addition, the selection of ERP system is not only based on overall average rating about an alternative and it is also based on the recent reviews about an alternative. Because, the average value represents the collected reviews over time do not always give the current customer review - recent reviews about the latest version are more likely to give the current sentiment. Therefore the selection should be based on both overall rating and latest version of an alternative. This can be presented by the amplitude and phase expression of CIFSS. The given each expert needs to select the appropriate ERP software system among the four alternatives $x_{1},$ $x_{2},$ $x_{3},$ and $x_{4}$. For this, the steps of the proposed method are implemented to find the desired alternative(s) as follows:

Step 1: The ratings of each expert towards the evaluation of alternatives under the different criteria are provided in terms of linguistic expressions as given in Table 2. In this table, each pair of linguistic intimation represents the amplitude and phase term. The amplitude term means to give an expert's decision with respect to alternative and the phase term can be used to denote an expert's decision with respect to the current version.

For instance, suppose the expert $E_{1}$ thinks that the over all rating about $x_{1}$ is "Low" and the rating about the current version of $x_{1}$ is "Slightly Low" for the first attribute. So the value of $x_{1}$ for the first attribute can be represented as "L, SL". However, the respective ratings of each linguistic term in the form of CIFSS features are summarized in Table 3.

Step 2: By taking the experts' $E_{1}, E_{2}, E_{3}$ weight as 0.253, 0.621, 0.126, which is provided by the company senior members, we aggregate the different expert preferences into the collective one using Eq. (9). The results corresponding to them are summarized in Table 4.

Step 3: Considering the different experts' preferences weight as given in Table 5 with respect to each criteria, we aggregate their preferences and their result is summarized in the last column of the Table 5.

By using the steps of the SWARA techniques, as mentioned in Step 3 of the proposed method, the values of $k_j$, $p_j$ and $q_j$ are summarized in Table 6. In this Table, the most significant criteria is ranked as first and the least significant criteria is ranked as last.

Hence, the criteria weights are computed as

Step 4: Use Eqs (13), (14), we compute the values of $\alpha_{i}^{-}$, $\alpha_{i}^{+}$ for each alternative $x_i, i = 1, 2, 3, 4$ as

Step 5: Using the values of $\alpha_i^+, \alpha_i^-$, we compute the priority values $\mathbb{Q}_{i}$'s for each $x_{i}$ with Eq (15) and get

Step 6: The utility degree $\gamma_i$'s for each alternative is computed by using Eq (16) and get

From these values, the order preference of the given ERP software system is found as $x_{1}\succ x_{3}\succ x_{2} \succ x_{4}$ and, thus $x_{1}$ is the best alternative.

4.2.   Comparative analysis and advantages

In this section, the advantages of the proposed CIFS-SWARA-COPRAS method are underlined, which are listed as follows.

The criteria weights which are determined by SWARA approach include the exactness of experts' opinion in ERP software selection process. Also, COPRAS procedure contains the ratios of the benefit and the cost solutions, simultaneously. The automotive manufacturing company needs to select the suitable alternative of ERP system with its current version simultaneously. This two-dimensional problem can be solved by CIFSS atmosphere. In CIFSS, the amplitude term denoted an experts' decision regarding option of ERP system and the phase term denoted an experts' decision regarding current version of ERP system. This cannot be done by using traditional IFSs. To verify the strength of the proposed technique a comparative study is obtained in Table 7. From Table 7, we determined that $x_{3}$ is the best alternative by using ^[20,21]. This result is based on overall rating about an alternative. Suppose the rating about current version of an ERP software is taken into account, the result may be affected. This findings certify that the proposed technique is more powerful than previously developed techniques ^[6,20,21].

4.3.   Sensitivity analysis

In this section, we investigate a sensitivity according to the strategy value of the decision expert (DE) in [0, 1] by using Eq (17), which can be demonstrated by Eq (15).

where $\vartheta\in[0, 1]$. If $\vartheta < 0.5$, the DE indicates pessimistic type, (i.e., ) the least value is related as the weight to the benefit criteria. Consequently, if $\vartheta > 0.5$, the DE gives optimistic type and so, the least value is related as the weight to the cost criteria. Also, if $\vartheta = 0.5$, the DE indicates neutral and the same preference is associated with both benefit and cost criteria. From this, the various values of $\vartheta$ can able to evaluate the sensitivity of the proposed technique. The ranking values as to the parameters are given in Table 8 and Figure 2. From this Table, we can see that an alternative $x_{1}$ has the highest value, when $\vartheta =$ 0.5 to 1 whereas $x_{2}$ has the highest value when $\vartheta$ = 0 to 0.4. On the other side, $x_{4}$ has the lowest value when $\vartheta$ = 0 to 0.6 and $x_{2}$ has the lowest value when $\vartheta$ = 0.7 to 1. It is cleared that the proposed technique has good stability.

5.   Conclusions

The main contribution of the work is listed in the following points.

1) The present study proposed an integrated CIFS-SWARA-COPRAS method by utilizing the features of the CIFS information. CIFS is a poweful way to handle the imprecise information using the two-dimensional information including the phase term. In this stated framework, criteria weights are evaluated by the SWARA technique, and the ranking of alternatives is determined by the COPRAS method using CIFSs. The presented approach has been applied to handle the ERP software selection problem.

2) A multiple experts has been taken during the process to rate the information in terms of the CIFS features. A weighted average CIFWA operator has been used to aggregate such preferences. The criteria weight are computed by following the SWARA method which include the exactness of experts' opinion during the process.

3) A utility degree has been used to rank the different alternatives.

4) To verify the strength of the proposed technique a comparative study is conducted with the existing studies ^[6,20,21] and found that the proposed method has its superiority over these existing methods. Furthermore, the sensitivity analysis by varying the degree $\vartheta\in[0, 1]$. A decision maker may chose the the parameter according to their choices as pessimistic $\vartheta < 0.5$ or optimistic $\vartheta > 0.5$ or neutral $\vartheta = 0.5$ towards the benefit and cost criteria. The ranking order corresponding to each parameter is listed in Table 8 and Figure 2.

In future, the work will be developed, by considering a maximum number of alternatives and DEs as well as to solve real data. Moreover, we expand our work using different multi-criteria decision making platforms under the different environment ^[5,8,31,33].

Conflict of interest

There is no conflict of interest.

Acknowledgments

The authors would like to acknowledge to Consejo Nacional de Ciencia y Tecnologia for providing the funding with grant number CONACYT_SNI-CVU: 405262.