ERP selection using picture fuzzy CODAS method

Abstract

The purpose of this study is to develop an extension of CODAS method using picture fuzzy sets. In this respect, a new methodology is introduced to figure out how picture fuzzy numbers can be applied to CODAS method. COmbinative Distance-based Assessment (CODAS) is a new MCDM method proposed by Ghorabaee et al. Picture fuzzy sets (PFSs) are a new extension of ordinary fuzzy sets for representing human’s judgments having possibility more than two answers such as yes, no, refusal and neutral. Compared with other studies, the proposed method integrates multi-criteria decision analysis with picture fuzzy uncertainty based on Euclidean and Taxicab distances and negative ideal solution. ERP system selection problem is handled as the application area of the developed method, picture fuzzy CODAS. Results indicate that the new proposed method finds meaningful rankings through picture fuzzy sets. Comparative analyzes show that the presented method gives successful and robust results for the solutions of MCDM problems under fuzziness.

Keywords

Fuzzy picture fuzzy sets CODAS method ERP selection

1 Introduction

Fuzzy sets theory began with Zadeh’s presentation in the middle of 1960 s [1] with the purpose of explaining the uncertainty of entities by assigning membership degrees ranging from 0 to 1. It emerged as a new concept presenting the vagueness in real life problems for all researchers. However, fuzzy sets theory had some diffuculties to express all uncertainty in the real life problem because of the non-membership degree [2]. Therefore, after introducing fuzzy sets theory, many extensions of fuzzy sets have been improved by several researchers.

First extension of fuzzy sets was presented by Atanassov [3] with the idea of intuitionistic fuzzy sets (IFSs). It was introduced as a generalized version of fuzzy set by using membership and non-membership degrees. Atanassov and Gargov [4] and Atanassov [5] presented interval-valued intuitionistic fuzzy sets (IVIFS) that consist of a membership function, a non-membership function, and a hesitancy function. Yager [6] developed Pythagorean fuzzy sets (PyFS) and Wang and Li [7] integrated power Bonferroni mean operator to Pythagorean fuzzy numbers. Lately, Cuong and Kreinovich [8] suggested picture fuzzy sets (PFSs) and explained the principal processings and specifications of PFSs. In real applications, it has been observed that some cases cannot be represented in IFS and Pythagorean fuzzy set (PyFS). For example, considering a voting system, human views contain many such responses: yes, no, abstain and refusal [9]. The picture fuzzy set is constituted with three functions to clarify the degree of membership, the degree of neutral membership and the degree of non-membership. As the only limitation, it is indicated that the sum of three degrees should not pass 1 at all [10]. Cuong [11] introduced the concept of picture fuzzy sets (PFSs) that was the answer for the human’s ideas consisting more than two answers like, yes, no, refusal and neutral. Classical voting system is an illuminative example of this idea because the voters can be grouped into four categories such as vote for, vote neutral, vote against and vote refusal [12].

Many researchers have contributed to the implementation of PFS with the models they have developed. Singh [13] proposed the geometrical interpretation of picture fuzzy sets and correlation coefficients for picture fuzzy sets. Furthermore, Son [14] proposed a novel distributed picture fuzzy clustering method on PFSs. The model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis is prepared by Thong [15]. Cuong and Hai [16] made a definition as first for fuzzy logic operators and presented important operations for fuzzy derivation forms in the Picture fuzzy logic. Moreover, Cuong et al. [17] studied the representative of picture fuzzy t-norm and t-conorm. Phong et al. [18] presented specific configuration of picture fuzzy relations.

Thong and Son [19] presented Automatic Image Fuzzy Clustering for picture fuzzy sets (AFC-PFS) to accomplish the appropriate number of groups for fuzzy clustering of PFS. Wei [20] presented the multi attribute decision making (MADM) method based on the picture fuzzy cross entropy model proposed in the article. Garg [21] proposed a model on picture fuzzy sets and used it to multiple criteria decision making (MCDM) problems. Son [22] presented the generalized picture distance measures and picture association measures. Moreover, Son and Thong [23] developed some combined prediction models by using picture fuzzy clustering for weather update from satellite image series. Wei [24] presented the TODIM approach for picture fuzzy multiple attribute decision making problems. In 2018, Jana et al. [25] proposed some aggregation operators that is named as Dombi operators for PFSs conditions and applied these actions to MADM process. Wei and Gao [26] proposed the common dice likeness surveys for PFSs. Wei [27] presented picture fuzzy Hamacher collection operators with traditional Hamacher processes at MADM. Wei et al. [28] presented the projection models by using picture fuzzy information for multi attribute decision making problems. Wei et al. [29] defined picture 2-tuple linguistic operators in MADM. Wei [30] also defined some picture uncertain linguistic Bonferroni mean operators for MADM.

Ashraf et al. [31] proposed a model of cubic picture fuzzy sets that was the extension form of picture fuzzy sets. In 2019, Ashraf et al. [32] improved a new approach on geometric aggregation operators and TOPSIS method to understand the ambiguity in decision making problems for picture fuzzy sets. Zeng et al. [33] explored the exponential Jensen Picture fuzzy divergence measure and examined its implementations in multi-criteria decision making (MCDM) problems. Moreover, applications to PFSs can be accomplish to many real world cases like decision-making problems [20, 34], shareholder voting [35], geographic data clustering [36] and weather nowcasting [23].

The motivation of our paper is to integrate the superiorities of CODAS method and picture fuzzy sets. CODAS method’s superiority is that it is based on not only Euclidean distance but also Taxicab distance. Decisions should not be based on only direct distances between ideal solutions and alternatives but also indirect distances. On the other hand, picture fuzzy sets present a possibility for decision makers to express their hesitancy and refusal degrees at the same time. Therefore, we develop a new methodology to overcome MCDM problems, which is based upon CODAS method under picture fuzzy environment. CODAS (Combinative Distance-based ASsessment) method was first presented by Ghorabaee et al. [37]. By considering CODAS method many advantages can be realized that have not considered in other multi criteria decision making (MCDM) methods. The advantage of the CODAS method over other distance-based decision-making methods is that it is based on two types of distances. Using two types of distances in the assessment process leads to improve the accuracy of the ranking results [38].

To show the validity of the proposed picture fuzzy CODAS (PF-CODAS) method, an application to enterprise resource planning (ERP) system problem (adopted from [39]) is presented.

Due to the difficulties and constraints on business organizations, ERP system selection is critical and time consuming process. Nevertheless, the importance of selecting an appropriate ERP system should not be underlined, given the significant financial investments and potential risks and benefits [40]. Selection of the suitable ERP system is important for the successful implementation of ERP projects. Therefore, in this study, a method for removing the ambiguity in linguistic evaluations has been developed for the ERP system selection problem.

The remainder of the paper is organized as follows. In Section 2 literature review on CODAS method and current literature on ERP selection problems are given. In Section 3 the basic concepts of PFS and the steps of the crisp CODAS method are proposed. In Sections 4 and 5, the steps of the proposed PF-CODAS method and an explanatory numerical example are presented, respectively. Finally, a comparative analysis is presented in Section 6 and the study is concluded in Section 7.

2 Literature review

Because of its approach to MCDM problems, CODAS method has been often employed in the solution of numerous problems in the literature. CODAS method has been extended by almost all of the fuzzy set types except picture fuzzy sets. The integration of picture fuzzy sets and CODAS will fill a gap that has not yet been addressed in the literature. Within the scope of literature research, “CODAS” and “ERP Selection” keywords are searched on the web of science database and a brief analysis is done about the results. These articles are summarized in this section.

2.1 Literature review on CODAS method

The CODAS method was firstly presented by Ghorabaee et al. [37] to solve complex MCDM problems. In this study, the results of the CODAS method were compared with some of the actual MCDM methods and as a result, it has been revealed that the CODAS method is an effective application to deal with MCDM problems [41]. Ghorabaee et al. [38] also presented a combined model that contains the fuzzy logic and CODAS method for multi-criteria group decision-making problems to get the best suppliers. In order to develop CODAS method, linguistic terms and corresponding trapezoidal fuzzy numbers (TrFNs) were included.

Panchal et al. [42] proposed an integrated MCDM method for examining the maintenance decision problem in a process industry, and the basis of this method is the fuzzy analytical hierarchy process (FAHP) and a current fuzzy CODAS method. Badi et al. used CODAS method for location selection of desalination facilities [43] and for supplier selection in the steelmaking company [44]. Boltürk [45] presented the pythagorean fuzzy extension of CODAS method, namely Pythagorean fuzzy CODAS, for the problem of selection the best supplier. Later, Boltürk and Kahraman [46] developed an intuitionistic fuzzy CODAS method, called Interval-Valued Intuitionistic Fuzzy CODAS, to select wave energy facility location and the obtained results were compared with crisp CODAS method’s. As a result of the study, a different ranking was obtained by stating the reason for new ranking clearly. Mathew and Sahu [47] solved two material handling equipment (conveyor and automated guided vehicle) selection problem using various newly developed CODAS, EDAS, WASPAS and MOORA methods. The rankings resulted with these four approaches were compared with other popular approaches like TOPSIS and ELECTRE.

Peng and Garg [48] proposed weighted distance based approximation (WDBA), CODAS and a new method for constructing of distance measure and similarity measure to handle interval-valued fuzzy soft decision-making problems. Pamucar et al. [49] introduced Pairwise-CODAS model in which modification of the CODAS method was carried out using Linguistic Neutrosophic Numbers (LNN). Yeni and Ozcelik [50] presented interval-valued Atanassov intuitionistic fuzzy CODAS (IVAIF-CODAS) method for the solution of the MCGDM problem. Adalı and Tuş [51] presented a model including CRITIC to calculate the objective weights and TOPSIS, EDAS, CODAS methods to measure performances of hospital site alternatives.

Interval-valued neutrosophic CODAS method was developed by Karaşan et al. [52] for the selection problem among wind energy plant locations. Ijadi Maghsoodi et al. used hybrid decision-making approaches for a material selection problem [53] by applying Step-Wise Weight Assessment Ratio Analysis (SWARA) method and CODAS and for a site selection problem [54] by applying Best-Worst Method (BWM) method and CODAS. Interval-valued intuitionistic trapezoidal fuzzy sets (IVITrFS) and CODAS method are the basis of a new integrated model proposed by Seker [41]. In this study, a different ranking was obtained and superiority of the new method was explained.

Studies in the literature indicate that various fuzzy number extensions considered under uncertainty are integrated with the CODAS method. In this respect, picture fuzzy extension of CODAS method is emphasized in this study which has not been discussed in the literature yet.

2.2 Current literature on ERP selection problems

Market competition between companies in international markets has significantly changed the business environment by decreasing total costs, increasing return on investments, shortening product delivery times and providing sensitivity to customer demands. In this direction, enterprises should use effective corporate information systems in order to increase their competitive advantage. ERP is becoming more critical due to its capability to combine material, finance, and information flow and support organizational strategies [55, 56]. Gable and Stewart [57] defines ERP as one of the best application for accomplishing competitive advantage for enterprises. By this way, it can be concluded that ERP selection process is crucial for ensuring competitiveness and agility of the companies.

All ERP applications have different possibilities. Therefore, any of the ERP application will not provide best solutions to enterprises [40 , 59]. It can be stated that examining flexibility of ERP systems will lead decision maker to choose the best ERP system between alternatives.

Many researches have been conducted about ERP selection in literature. Most of the studies have been leaded to explain the ERP selection process for larger enterprises [60 –62]. In other way, some studies have focused on small medium enterprises (SMEs) [63 –65]. Illa et al. [60] improved a new methodology for ERP selection which is called SHERPA. The mentioned methodology based on natural language definitions of the system domain, user needs and candidate ERP solutions. Badri et al. [66] proposed a 0–1 goal programming model to choose information system between alternatives according to multiple criteria including benefits, hardware, software and other costs, risk factors, preferences of decision makers and users, completion time, and training time constraints.

Lin and Ford [64] developed a road map for SMEs and classified the selection process in five steps. The steps are containing project initiation, business process reengineering, business requirements, creating a business case and the selection process. Yen and Sheu [67] presented competitive preferences as volume and flexibility. Defining ERP selection process as competitiveness, ERP packages should let simple information contribution, higher regional self goverment, easier log in to database and more software applications conformation.

Liao et al. [68] proposed a new model according to the 2-tuple linguistic information computing. Consistency and inconsistency indices have been defined taking into account the information acquired from internal interviews with ERP vendors. In addition, a linear programming model is created to select the most appropriate ERP system.

Vatansever and Uluköy [69] addressed the issue of determining the most suitable software for the manufacturing industry using fuzzy AHP and fuzzy MOORA methods respectively to determine the weight of criteria and to evaluate the alternatives.

A four-step multi-criteria decision making approach has been proposed in 2019 by Hinduja and Pandey [70]. Three MCDM methods, namely DEMATEL, IF-ANP and IF-AHP, were used at different stages of the said approach. Intuitionistic fuzzy sets were used to capture the uncertainties and hesitations in decision makers’ judgments.

3 Preliminaries

In this section, the preliminaries on picture fuzzy numbers and steps of the crisp CODAS methodology are presented.

3.1 Basic concepts of picture fuzzy sets

Definition 1. A picture fuzzy set (PFS) on the universe X is an object of the form A = {x, μ (x) , η (x) , ν (x) x ∈ X},

where μ_A (x) (ε [0, 1]) is named as the “degree of positive membership of A”, η_A (x) (ε [0, 1]) is named as the “degree of neutral membership of A” and v_A (u) (ε [0, 1]) is named as the “degree of negative membership of A”, and μA (x), ηA (x), νA (x) satisfy the following condition: 0 ≤ μ (x) + η (x) + ν (x) ≤1, ∀x ∈ X. Then for x ∈ X, π_A (x) = 1 - (μ (x) + η (x) + ν (x)) could be named as the degree of refusal membership of x in A [8].

Definition 2. Let A = (μ_A, η_A, ν_A) and B = (μ_B, η_B, ν_B) be two picture fuzzy numbers (PFNs), the operation formula is determined [8]: $A \oplus B = (μ_{A} + μ_{B} - μ_{A} μ_{B}, η_{A} η_{B}, v_{A} v_{B});$ (1)

$A \otimes B = (μ_{A} μ_{B}, η_{A} + η_{B} - η_{A} η_{B}, v_{A} + v_{B} - v_{A} v_{B});$ (2)

$λ A = (1 - {(1 - μ_{A})}^{λ}, η_{A}^{λ}, v_{A}^{λ}), λ > 0;$ (3)

$A^{λ} = (μ_{A}^{λ}, 1 - {(1 - η_{A})}^{λ}, 1 - {(1 - v_{A})}^{λ}), λ > 0;$ (4)Definition 3. Let A = (μ_A, η_A, ν_A) and B = (μ_B, η_B, ν_B) be PFNs, the score and accuracy functions of A and B are illustrated [8]: $S (A) = μ_{A} - v_{A}, S (B) = μ_{B} - v_{B};$ (5)

$H (A) = μ_{A} + η_{A} + v_{A}, H (B) = μ_{B} + η_{B} + v_{B};$ (6)

For two PFNs A and B, according to the Definition 3, then $if s (A) ≺ s (B), then A ≺ B;$ $if s (A) ≻ s (B), then A ≻ B;$ $if s (A) = s (B), h (A) ≺ h (B) then A ≺ B;$ $if s (A) = s (B), h (A) ≻ h (B) then A ≻ B;$ $if s (A) = s (B), h (A) = h (B) then A = B;$

Definition 4. The normalized picture fuzzy Euclidean distance [8]: $e_{p} (A, B) = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} (\begin{matrix} {(μ_{A} (x_{i}) - μ_{B} (x_{i}))}^{2} + \\ {(η_{A} (x_{i}) - η_{B} (x_{i}))}^{2} + \\ {(v_{A} (x_{i}) - v_{B} (x_{i}))}^{2} \end{matrix})};$ (7)

Definition 5. The weighted decision matrix T = (t_ij) _m*n can be obtained as follows [71]: $t_{ij} = w_{j} r_{ij} = 〈 1 - {(1 - μ_{r_{ij}})}^{w_{j}}, η_{r_{ij}}^{w_{j}}, {(η_{r_{ij}} + v_{r_{ij}})}^{w_{j}} - η_{r_{ij}}^{w_{j}} 〉;$ (8) where w_j represents the weight of jth criteria.

3.2 CODAS method

The CODAS method is a newly developed and one of the current MCDM method. In the method, the distance measurements taken as a basis for selecting the best alternative from the available options are Euclidean and Hamming. The primary idea of CODAS method can be stated as the best option should have the farthest distance from the negative ideal solution. When the Euclidean distances of two options have the same crisp number, Hamming distances are compared to choose the best one [37]. The steps of CODAS method are as follows [37 , 43]:

Step 1. Build decision matrix (X). Decision makers evaluate possible alternatives from the point of each qualification. ${[x_{ij}]}_{n x m} = [\begin{matrix} \begin{matrix} x_{11} & x_{12} \end{matrix} \begin{matrix} \dots & x_{1 m} \end{matrix} \\ \begin{matrix} x_{21} & x_{22} \end{matrix} \begin{matrix} \dots & x_{2 m} \end{matrix} \\ \begin{matrix} ⋮ & ⋮ ⋮ & ⋮ \end{matrix} \\ \begin{matrix} x_{n 1} & x_{n 2} \end{matrix} \begin{matrix} \dots & x_{nm} \end{matrix} \end{matrix}]$ (9) where x_ij (x_ij ≥ 0) shows the performance value of ith alternative based on jth criterion (i ∈ {1, 2, …, n} and j ∈ {1, 2, …, m}).

Step 2. Obtain a normalized decision matrix by normalizing the decision matrix according to the type of criterion. Linear normalization of performance values is used as follows, where N_b and N_c represent the sets of beneficial and non-beneficial criteria respectively: $n_{ij} = {\begin{matrix} \frac{x_{ij}}{max_{i} x_{ij}} if j \in N_{b} \\ \frac{min_{i} x_{ij}}{x_{ij}} if j \in N_{c} \end{matrix}$ (10)

Step 3. Obtain a weighted normalized decision matrix by weighting the normalized decision matrix. Equation 11 is used to calculate the weighted normalized performance values: $r_{ij} = w_{ij} * n_{ij}$ (11) where w_j (0 < w_j < 1) represents the weight of jth criterion, and $\sum_{j = 1}^{m} w_{j} = 1$ .

Step 4. Determine the negative-ideal solution (point) as follows: $ns = {[{ns}_{j}]}_{1 x m}$ (12)

${ns}_{j} = min r_{ij}$ (13)

Step 5. Compute the Euclidean (E_i) and Taxicab (T_i) distances of alternatives from the negative-ideal solution, shown as expressed in Equations 6-7: $E_{i} = \sqrt{\sum_{j = 1}^{m} {(r_{ij} - {ns}_{j})}^{2}}$ (14)

$T_{i} = \sum_{j = 1}^{m} | r_{ij} - {ns}_{j} |$ (15)

Step 6. Create the relative assessment (Ra) matrix, shown as follows: $Ra = {[h_{ik}]}_{n x n}$ (16)

$h_{ik} = (E_{i} - E_{k}) + (ψ (E_{i} - E_{k}) x (T_{i} - T_{k}))$ (17) where k ∈ {1, 2, …, n} and ψ is a threshold function to recognize the equality of the Euclidean distances of two alternatives, and is defined as follows: $ψ (x) = {\begin{matrix} 1 if | x | ⩾ τ \\ 0 if | x | < τ \end{matrix}$ (18)

The threshold parameter (τ) can be set by decision-maker and this parameter’s value is between 0.01 and 0.05. If the difference between Euclidean distances of two alternatives is less than τ, these two alternatives are also compared by the Taxicab distance.

Step 7. Calculate the assessment score (H_i) of each alternative, shown as follows: $H_{i} = \sum_{k = 1}^{n} h_{ik}$ (19)

The higher H_i, the most appropriate alternative.

Step 8. Rank the alternatives according to the decreasing values of H_i. The alternative with the highest H_i is the best choice among the alternatives.

4 Proposed picture fuzzy CODAS method

According to definitions of PFS and CODAS method, the steps of presented picture fuzzy CODAS method are listed below.

Step 1. Establish the picture fuzzy decision matrix as prepared in Equation (20). $\tilde{X} = {[{\tilde{x}}_{ij}]}_{n x m} = [(\begin{matrix} {\tilde{x}}_{11} {\tilde{x}}_{12} & \dots & {\tilde{x}}_{1 m} \\ ⋮ & ⋱ & ⋮ \\ {\tilde{x}}_{n 1} {\tilde{x}}_{n 2} & \dots & {\tilde{x}}_{nm} \end{matrix})]$ (20) where ${\tilde{x}}_{ij} ⩾ 0$ and ${\tilde{x}}_{ij} = {μ (x), η (x), ν (x)}, 0 ⩽ μ (x) + η (x) + ν (x) ⩽ 1$ .

Step 2. Calculate the picture fuzzy normalized matrix.

If the criteria is occurred in cost type, the formulation below is used for normalization of the criteria. There is no need to normalization for benefit criteria in picture fuzzy sets [9]. $r_{ij} = {v (x), η (x), μ (x)}$ (21)

Step 3. Compute the picture fuzzy weighted normalized matrix.

Picture fuzzy weighted normalized matrix values are calculated by using Equation (8) where w_j gets values between 0 and 1.

Step 4. Determine the negative-ideal solution (point) for picture fuzzy weighted normalized values.

Score functions and accuracy functions are used to identify negative ideal solutions for all criteria. To calculate score and accuracy functions, Equation (5) and Equation (6) are determined. The lowest scores are defined as negative ideal solutions.

Step 5. Calculate the Euclidean (E_i) and Taxicab (T_i) distances of alternatives from the negative-ideal solution.

In this step, Equation (7) is implemented for calculating Euclidean distances. For determining Taxicab distance, the normalized hamming distance formula is used illustrated in Cuong and Kreinovich [8] as below:

$\begin{matrix} d_{p} (A, B) = \frac{1}{n} \sum_{i = 1}^{n} (| μ_{A} (x_{i}) - μ_{B} (x_{i}) | \\ + | η_{A} (x_{i}) - η_{B} (x_{i}) | + | v_{A} (x_{i}) - v_{B} (x_{i}) |) \end{matrix}$ (22)

Step 6. Create the relative assessment (Ra) matrix with Equation (16) where k ∈ {1, 2, …, n} and ψ is a threshold function to identify the equivalence of the Euclidean distances of two alternatives, and is described in Equation (18).

The threshold parameter (τ) can be adjusted by decision-maker and this parameter’s value must be between 0.01 and 0.05. If the difference between Euclidean distances of two alternatives examined is less than τ, these two alternatives will be also calculated with the Taxicab distance.

Step 7. Calculate the assessment score (H_i) of each alternative with Equation (19). The higher H_i is chosen as best alternative.

Step 8. Rank the alternatives by decreasing values of H_i. In final, the alternative having the highest H_i is chosen as the best choice between all alternatives.

5 Application

In this section, the proposed Picture Fuzzy CODAS (PF-CODAS) method is applied to an example of enterprise resource planning (ERP) system problem which is adopted from [39]. Wei adapted this problem from Liao et al. [68]. There are five possible ERP systems A_t (t = 1, 2, ... 5) and four unquantifiable evaluation criteria C_i (i = 1, 2, 3, 4). These criteria are function and technology (C1), strategic fitness (C2), vendor’s ability (C3), vendor’s reputation (C4).

All the criteria are benefit-type in this study and the weights were determined by the decision-makers as follows respectively (0.2, 0.1, 0.3, 0.4).

Weights of the criteria for selected problem are known as a scope of this study taken from Wei [39]. There are few studies determining weights of the criteria in the literature. Zhang et al. [72] developed a new approach for the problem of two-sided matching decision making (TSMDM). In this study [72], the primary weight vector is obtained directly from “fuzzy preference relation with self-confidence” based on logarithmic least squares method. This approach also provides algorithms to improve consistency. For another study [73], an approach was developed with multi-granular hesitant fuzzy linguistic term sets for the TSMDM problem. Zhang et al. [73] created optimization models to assign criteria weights by not ensuring clear criteria weight vectors.

In order to determine the most suitable ERP system, the steps of the proposed PF-CODAS approach is presented as follows:

Step 1. Picture fuzzy decision matrix which is taken from ERP selection example is constructed in Table 1.

Table 1
Picture fuzzy decision matrix

C1 C2 C3 C4

A1 (0.53, 0.33, 0.09) (0.89, 0.08, 0.03) (0.42 0.35, 0.18) (0.08, 0.89, 0.02)

A2 (0.73, 0.12, 0.08) (0.13, 0.64, 0.21) (0.03, 0.82, 0.13) (0.73, 0.15, 0.08)

A3 (0.91, 0.03, 0.02) (0.07, 0.09, 0.05) (0.04, 0.85, 0.10) (0.68, 0.26, 0.06)

A4 (0.85, 0.09, 0.05) (0.74, 0.16, 0.10) (0.02, 0.89, 0.05) (0.08, 0.84, 0.06)

A5 (0.90, 0.05, 0.02) (0.68, 0.08, 0.21) (0.05, 0.87, 0.06) (0.13, 0.75, 0.09)

	C1	C2	C3	C4
A1	(0.53, 0.33, 0.09)	(0.89, 0.08, 0.03)	(0.42 0.35, 0.18)	(0.08, 0.89, 0.02)
A2	(0.73, 0.12, 0.08)	(0.13, 0.64, 0.21)	(0.03, 0.82, 0.13)	(0.73, 0.15, 0.08)
A3	(0.91, 0.03, 0.02)	(0.07, 0.09, 0.05)	(0.04, 0.85, 0.10)	(0.68, 0.26, 0.06)
A4	(0.85, 0.09, 0.05)	(0.74, 0.16, 0.10)	(0.02, 0.89, 0.05)	(0.08, 0.84, 0.06)
A5	(0.90, 0.05, 0.02)	(0.68, 0.08, 0.21)	(0.05, 0.87, 0.06)	(0.13, 0.75, 0.09)

Step 2. Normalized picture fuzzy decision matrix is established in Table 2. In numerical example examined, all criteria are beneficial so that normalized picture fuzzy decision matrix will be same with picture fuzzy decision matrix. Therefore, the matrix presented in Table 1 is also the normalized picture fuzzy decision matrix.

Table 2

Weighted normalized picture fuzzy decision matrix and distances

	C1	C2	C3	C4	E_i	T_i
A1	(0.14, 0.80, 0.04)	(0.20, 0.78, 0.03)	(0.15, 0.73, 0.10)	(0.03, 0.95, 0.01)	0.17	0.17
A2	(0.23, 0.65, 0.07)	(0.01, 0.96, 0.03)	(0.01, 0.94, 0.04)	(0.41, 0.47, 0.09)	0.29	0.25
A3	(0.38, 0.50, 0.05)	(0.01, 0.79, 0.04)	(0.01, 0.95, 0.03)	(0.37, 0.58, 0.05)	0.29	0.30
A4	(0.32, 0.62, 0.06)	(0.13, 0.83, 0.04)	(0.01, 0.97, 0.02)	(0.02, 0.95, 0.03)	0.14	0.13
A5	(0.37, 0.55, 0.04)	(0.11, 0.78, 0.11)	(0.02, 0.96, 0.02)	(0.05, 0.89, 0.04)	0.18	0.19
NIS	(0.14, 0.80, 0.04)	(0.01, 0.96, 0.03)	(0.01, 0.97, 0.02)	(0.02, 0.95, 0.03)

Step 3. Weighted normalized picture fuzzy decision matrix is obtained in Table 2. By using Equation (8), weights of the criteria (0.2, 0.1, 0.3, 0.4) are multiplied with normalized picture fuzzy decision matrix.

Step 4. Negative ideal solutions (NIS) are also identified in Table 2. To obtain negative ideal solution, score functions are calculated. For C1, score functions are calculated for each alternative. For A1, it is calculated as; 0.14-0.80-0.04=–0.701. By using the same formulas, score functions for A2, A3, A4 and A5 are calculated –0.494, –0.167, –0.359 and –0.218 respectively. Score function of A1 is obtained in lowest score, so it is chosen as negative ideal solution.

Step 5. Euclidean (E_i) and Taxicab (T_i) distances of alternatives from the negative-ideal solution is also obtained in Table 2 by calculating Equations (7) and (22).

Step 6. Relative assessment (Ra) matrix is constructed by using (E_i - E_k) and (T_i - T_k) values in Table 3. To calculate A1-A2 value in relative assessment matrix, E1-E2 (0.17–0.29=–0.12) values and T1-T2 (0.17–0.25=–0.08) values are added (–0.12+(–0.08)= –0.19).

Table 3

Relative assessment matrix

	A1	A2	A3	A4	A5	H _i
A1	0,00	–0,19	–0,25	0,08	–0,01	–0,37
A2	0,19	0,00	–0,01	0,28	0,16	0,62
A3	0,25	0,01	0,00	0,34	0,22	0,82
A4	–0,08	–0,28	–0,34	0,00	–0,11	–0,81
A5	0,01	–0,16	–0,22	0,11	0,00	–0,26

Step 7. Assessment scores (H_i) are calculated by summation of all scores. Alternative that has the highest assessment score is chosen as best alternative. Hi scores are listed in Table 3.

Step 8. According to assessment scores, ranking of the alternatives is obtained as A3 > A2 > A5 > A1 > A4 in final step of PF-CODAS method.

6 Comparative analysis

In order to check the validity of the proposed PF-CODAS method in this study, example application is adopted from Wei [39]. In this context, the results are compared with the existing methods of PF-TOPSIS [74] and picture fuzzy aggregation operators [39]. PF-TOPSIS method is selected due to the similarity to the proposed PF-CODAS. Due to the application of Wei’s example, the results obtained in this study are also compared with the results of picture fuzzy aggregation operators as picture fuzzy weighted average (PFWA) and picture fuzzy weighted geometric (PFWG). The same weights of criteria are considered by these methods. The comparative results are shown in Table 4.

Table 4
Comparison of proposed method with the existing methods

A1 A2 A3 A4 A5 Order

PF-CODAS (proposed method) –0,37 0,62 0,82 –0,81 –0,26 A3 > A2 > A5 > A1 > A4

PF-TOPSIS [74] 0,295 0,293 0,322 0,231 0,261 A3 > A1 > A2 > A5 > A4

PFWA [39] 0,350 0,394 0,504 0,329 0,349 A3 > A2 > A1 > A5 > A4

PFWG [39] 0,174 0,140 0,185 0,055 0,093 A3 > A1 > A2 > A5 > A4

	A1	A2	A3	A4	A5	Order
PF-CODAS (proposed method)	–0,37	0,62	0,82	–0,81	–0,26	A3 > A2 > A5 > A1 > A4
PF-TOPSIS [74]	0,295	0,293	0,322	0,231	0,261	A3 > A1 > A2 > A5 > A4
PFWA [39]	0,350	0,394	0,504	0,329	0,349	A3 > A2 > A1 > A5 > A4
PFWG [39]	0,174	0,140	0,185	0,055	0,093	A3 > A1 > A2 > A5 > A4

The preference order obtained by the proposed method and PF-TOPSIS [74], PFWA, PFWG operators provided by Wei [39] are slightly different but the desired best alternative is same. In all four methods, A3 is the best alternative, and A4 is the worst. According to the Table 5, the results of each method show A3 as the best alternative and A4 as the worst.

Table 5

Ranking similarity for four methods

	PF-CODAS (proposed method)	PF-TOPSIS [74]	PFWA [39]	PFWG [39]
PF-CODAS (proposed method)	1 (100%)	–0,3 (40%)	–0,6 (60%)	–0,3 (40%)
PF-TOPSIS [74]	–0,3 (40%)	1 (100%)	0,9 (60%)	1 (100%)
PFWA [39]	–0,6 (60%)	0,9 (60%)	1 (100%)	0,9 (60%)
PFWG [39]	–0,3 (40%)	1 (100%)	0,9 (60%)	1 (100%)

The Spearman’s rank correlation coefficient values (rs) are calculated for the comparative analysis of the methods (Table 5). The values in parentheses represent the correspondence for the ranked alternatives between the PF-CODAS method and other three methods as calculated by [75]. For example, the value of 40% at the intersection of the first row (PF-CODAS) and the second column (PF-TOPSIS) was obtained due to the fact that two of the five alternatives (A3 and A4) were in the same order according to the order of the two methods given in Table 4.

Based on the values of Spearman correlation coefficient, the ranks provided by the PF-CODAS method seem very similar to the PFWA method and a 60% correspondence is observed between these two methods. Also, PF-CODAS and PF-TOPSIS (and PFWG) are similar according to observed 40% correspondence [75].

7 Conclusion

In this paper, picture fuzzy CODAS method is presented based on traditional CODAS method. First of all, picture fuzzy sets and its basic definitions are briefly shown including Euclidean distance, Taxicab distance, score functions and accuracy functions. After definitions, extension of picture fuzzy sets with traditional CODAS method is proposed by methodology of the study. By using picture fuzzy numbers, decision makers can report their agreement, disagreement and hesitancy degrees and add them into decision making process. Finally, a numerical example for ERP selection problem has been given to express the new model and make comparisons between other picture fuzzy methods. As a result of numerical example, we state that Picture fuzzy CODAS method give meaningful results and rankings under fuzziness.

In order to present the validity of the proposed method in this study, statistical analysis is also used in comparison with other methods. Statistical values also indicate that the method presented gives valid results. Considering the difficulty of making decisions under uncertainty, the new method developed in our study will provide effective results in decision-making processes. Compared with previous methods, one of the advantage of our proposed method is that it is based on the negative ideal distance measure with lower processing complexity.

However, the results of this study are limited with ERP selection problem. In future, PF-CODAS method can be applied to many other decision-making problems such as machine selection or vehicle selection problems by management.

References

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–356.

Qiyas

, Abdullah

, Ashraf

, Khan

and Khan

, Triangular picture fuzzy linguistic induced ordered weighted aggregation operators and its application on decision making problems, Mathematical Foundations of Computing 2(3) (2019), 183–201.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), 87–96.

Atanassov

K.T.

and Gargov

, Interval valued intuitionistic fuzzy-sets, Fuzzy Sets and Systems 31 (1989), 343–349.

Atanassov

K.T.

, Operators over interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 64 (1994), 159–174.

Yager

R.R.

, Pythagorean fuzzy subsets, Proceedings of Joint IFSA World Congress and NAFIPS Annual Meeting, Edmonton, IEEE, 2013, 57–61.

Wang

and Li

, Pythagorean fuzzy interaction power Bonferroni mean aggregation operators in multiple attribute decision making, International Journal of Intelligent Systems 35(1) (2020), 150–183.

Cuong

B.C.

and Kreinovich

, Picture Fuzzy Sets – a new concept for computational intelligence problems, Proceedings of 2013 Third World Congress on Information and Communication Technologies, IEEE, 2013, 1–6.

Khan

A.A.

, Ashraf

, Abdullah

, Qiyas

, Luo

and Khan

S.U.

, Pythagorean fuzzy Dombi aggregation operators and their application in decision support system, Symmetry 11 (2019), 383.

10.

Zhang

, Zhan

, Wu

W.Z.

and Alcantud

J.C.R.

, Fuzzy β-covering based-fuzzy rough set models and applications to multi-attribute decision making, Computers and Industrial Engineering 128 (2019), 605–621.

11.

Cuong

, Picture fuzzy sets-first results. part 1, in: Seminar “Neuro-Fuzzy Systems with Applications”, Institute of Mathematics Hanoi, 2013.

12.

Akram

, Habib

and Koam

A.N.

, A novel description on edge-regular q-rung picture fuzzy graphs with application, Symmetry 11(4) (2019), 489.

13.

Singh

, Correlation coefficients for picture fuzzy sets,, Journal of Intelligent & Fuzzy Systems 28 (2015), 591–604.

14.

Son

L.H.

, DPFCM: A novel distributed picture fuzzy clustering method on picture fuzzy sets, Expert Systems with Applications 42 (2015), 51–66.

15.

Thong

N.T.

, HIFCF: An effective hybrid model between picture fuzzy clustering and intuitionistic fuzzy recommender systems for medical diagnosis, Expert Systems with Applications 42(7) (2015), 3682–3701.

16.

Cuong

B.C.

and Hai

P.V.

, Some fuzzy logic operators for picture fuzzy sets, In Knowledge and Systems Engineering (KSE), 2015, 132–137.

17.

Cuong

B.C.

, Kreinovitch

and Ngan

R.T.

, A classification of representable t-norm operators for picture fuzzy sets, In Knowledge and Systems Engineering (KSE), 2016, 19–24.

18.

Phong

P.H.

, Hieu

D.T.

, Ngan

R.T.

and Them

P.T.

, Some compositions of picture fuzzy relations, Proceedings of the 7th National Conference on Fundamental and Applied Information Technology Research (FAIR?), Thai Nguye, 2014, 19–40.

19.

Thong

P.H.

and Son

L.H.

, A novel automatic picture fuzzy clustering method based on particle swarm optimization and picture composite cardinality, Knowledge-Based Systems 109 (2016), 48–60.

20.

Wei

G.W.

, Picture fuzzy cross-entropy for multiple attribute decision making problems, Journal of Business Economics and Management 17 (2016), 491–502.

21.

Garg

, Some picture fuzzy aggregation operators and their applications to multicriteria decision-making, Arabian Journal for Science and Engineering 42(12) (2017), 5275–5290.

22.

Son

L.H.

, Measuring analogousness in picture fuzzy sets: From picture distance measures to picture association measures, Fuzzy Optimization and Decision Making 16 (2017), 359–378.

23.

Son

L.H.

and Thong

P.H.

, Some novel hybrid forecast methods based on picture fuzzy clustering for weather nowcasting from satellite image sequences, Applied Intelligence 46 (2017), 1–15.

24.

Wei

G.W.

, TODIM method for picture fuzzy multiple attribute decision making, Informatica 29 (2018), 555–566.

25.

Jana

, Senapati

, Pal

and Yager

R.R.

, Picture fuzzy Dombi aggregation operators: Application to MADM process, Applied Soft Computing 74 (2018), 99–109.

26.

Wei

G.W.

and Gao

, The generalized dice similarity measures for picture fuzzy sets and their applications, Informatica 29 (2018), 107–124.

27.

Wei

G.W.

, Picture fuzzy hamacher aggregation operators and their application to multiple attribute decision making, Fundamenta Informaticae 157 (2018), 271–320.

28.

Wei

G.W.

, Alsaadi

F.E.

, Hayat

and Alsaedi

, Projection models for multiple attribute decision making with picture fuzzy information, International Journal of Machine Learning and Cybernetics 9 (2018), 713–719.

29.

Wei

G.W.

, Alsaadi

F.E.

, Hayat

and Alsaedi

, Picture 2-tuple linguistic aggregation operators in multiple attribute decision making, Soft Computing 22 (2018), 989–1002.

30.

Wei

G.W.

, Picture uncertain linguistic Bonferroni mean operators and their application to multiple attribute decision making, Kybernetes 46 (2017), 1777–1800.

31.

Ashraf

, Abdullah

and Qadir

, Novel concept of cubic picture fuzzy set, Journal of New Theory 24 (2018), 59–72.

32.

Ashraf

, Mahmood

, Abdullah

and Khan

, Different approaches to multi-criteria group decision making problems for picture fuzzy environment, Bulletin of the Brazilian Mathematical Society, New Series 50(2) (2019), 373–397.

33.

Zeng

, Ashraf

, Arif

and Abdullah

, Application of exponential jensen picture fuzzy divergence measure in multi-criteria group decision making, Mathematics 7 (2019), 191.

34.

Wei

, Alsaadi

F.E.

, Hayat

and Alsaedi

, Projection models for multiple attribute decision making with picture fuzzy information, International Journal of Machine Learning and Cybernetics 9(4) (2018), 713–719.

35.

Nie

R.X.

, Wang

J.Q.

and Li

, A shareholder voting method for proxy advisory firm selection based on 2-tuple linguistic picture preference relation, Applied Soft Computing 60 (2017), 520–539.

36.

Hoa

N.D.

and Thong

P.H.

, Some improvements of fuzzy clustering algorithms using picture fuzzy sets and applications for geographic data clustering, VNU Journal of Science: Computer Science and Communication Engineering 32(3) (2017).

37.

Ghorabaee

K.M.

, Zavadskas

E.K.

, Turskis

and Antucheviciene

, A new combinative distance-based assessment (CODAS) method for multi-criteria decision-making, Economic Computation & Economic Cybernetics Studies & Research 50(3) (2016).

38.

Ghorabaee

K.M.

, Amiri

, Zavadskas

E.K.

, Hooshmand

and Antucheviciene

, Fuzzy extension of the CODAS method for multi-criteria market segment evaluation, Journal of Business Economics and Management 18(1) (2017), 1–19.

39.

Wei

, Picture fuzzy aggregation operators and their application to multiple attribute decision making,, Journal of Intelligent & Fuzzy Systems 33(2) (2017), 713–724.

40.

Teltumbde

, A framework of evaluating ERP projects, International Journal of Production Research 38 (2000), 4507–4520.

41.

Seker

, A novel interval-valued intuitionistic trapezoidal fuzzy combinative distance-based assessment (CODAS) method, Soft Computing 24(3) (2020), 2287–2300.

42.

Panchal

, Chatterjee

, Shukla

R.K.

, Choudhury

and Tamosaitiene

, Integrated fuzzy AHP-CODAS framework for maintenance decision in urea fertilizer industry, Economic Computation & Economic Cybernetics Studies & Research 51(3) (2017).

43.

Badi , Ballem

and Shetwan

, Site selection of desalination plant in Libya by using combinative distance-based assessment (CODAS) method, International Journal for Quality Research 12(3) (2018).

44.

Badi , Abdulshahed

A.M.

and Shetwan

, A case study of supplier selection for a steelmaking company in Libya by using the Combinative Distance-based ASsessment (CODAS) model, Decision Making: Applications in Management and Engineering 1(1) (2018), 1–12.

45.

Bolturk

, Pythagorean fuzzy CODAS and its application to supplier selection in a manufacturing firm, Journal of Enterprise Information Management 31(4) (2018), 550–564.

46.

Bolturk

and Kahraman

, Interval-valued intuitionistic fuzzy CODAS method and its application to wave energy facility location selection problem,, Journal of Intelligent & Fuzzy Systems 35(4) (2018), 4865–4877.

47.

Mathew

and Sahu

, Comparison of new multi-criteria decision making methods for material handling equipment selection, Management Science Letters 8(3) (2018), 139–150.

48.

Peng

and Garg

, Algorithms for interval-valued fuzzy soft sets in emergency decision making based on WDBA and CODAS with new information measure,, Computers & Industrial Engineering 119 (2018), 439–452.

49.

Pamučar

, Badi

, Sanja

and Obradović

, A novel approach for the selection of power-generation technology using a linguistic neutrosophic CODAS method: A case study in Libya, Energies 11(9) (2018), 2489.

50.

Yeni

F.B.

and Özçelik

, Interval-valued Atanassov intuitionistic Fuzzy CODAS method for multi criteria group decision making problems, Group Decision and Negotiation 28(2) (2019), 433–452.

51.

Adalı

E.A.

and Tuş

, Hospital site selection with distance-based multi-criteria decision-making methods, International Journal of Healthcare Management, 2019, 1–11.

52.

Karaşan

, Boltürk

and Kahraman

, A novel neutrosophic CODAS method: selection among wind energy plant locations,, Journal of Intelligent & Fuzzy Systems 36(2) (2019), 1491–1504.

53.

Ijadi Maghsoodi

, Poursoltan

, Antucheviciene

and Turskis

, Dam construction material selection by implementing the integrated SWARA-CODAS approach with target-based attributes, Archives of Civil and Mechanical Engineering 19 (2019), 1194–1210.

54.

Ijadi Maghsoodi

, Rasoulipanah

, López

L.M.

, Liao

and Zavadskas

E.K.

, Integrating interval-valued multi-granular 2-tuple linguistic BWM-CODAS approach with target-based attributes: Site selection for a construction project,, Computers & Industrial Engineering 139 (2020), 106–147.

55.

Yusuf

, Gunasekaran

and Abthorpe

M.S.

, Enterprise information systems project implementation: A case study of ERP in Rolls-Royce, Journal of Production Economics 87 (2004), 251–266.

56.

Yao

and He

H.C.

, Data warehousing and the internet’s impact on ERP, IT Professional 2(2) (2000), 37–41.

57.

Gable

and Stewart

, SAP R/3 implementation issues for small to medium enterprises, AMCIS Proceedings, 1999, 269.

58.

Sarkis

and Sundarraj

R.P.

, Factors for strategic evaluation of enterprise information technologies,, International Journal of Physical Distribution & Logistics Management 30(3-4) (2000), 196–220.

59.

Hong

K.K.

and Kim

Y.G.

, The critical success factors for ERP implementation: An organizational fit perspective,, Information & Management 40 (2002), 25–40.

60.

Illa

X.B.

, Franch

and Pastor

J.A.

, Formalising ERP selection criteria, Proceedings of the 10th International Workshop on Software Specification and Design, IEEE Computer Society, 2000, 115.

61.

Langenwalter

G.A.

, Enterprise Resources Planning and beyond: Integrating your entire organization, St Lucie Press, Boca Raton, FL, 2000.

62.

Verville

J.C.

and Halingten

, A qualitative study of the influencing factors on the decision process for acquiring ERP software, Qualitative Market Research: An International Journal 5(3) (2002), 188–198.

63.

Baki

and Çakar

, Determining the ERP package-selecting criteria: The case of Turkish manufacturing companies, Business Process Management Journal 11(1) (2005), 75–86.

64.

Lin and Ford

D.W.

, ERP selection for small and medium-sized manufactures, Proceedings International Conference of Manufacturing Research, 2004.

65.

Wei

, Chien

and Wang

M.J.

, An AHP-based approach to ERP system selection, International Journal of Production Economics 96(1) (2005), 47–62.

66.

Badri

M.A

, Davis

and Davis

, A comprehensive 0–1 goal programming model for project selection, International Journal of Project Management 19 (2001), 243–252.

67.

Yen

H.R.

and Sheu

, Aligning ERP implementation with competitive priorities of manufacturing firms: An exploratory study, International Journal of Production Economics 92(3) (2004), 207–220.

68.

Liao

X.W.

, Li

and Lu

, A model for selecting an ERP system based on linguistic information processing, Information Systems 32(7) (2007), 1005–1017.

69.

Vatansever

and Uluköy

, Determining enterprise resource planning systems through fuzzy AHP and fuzzy MOORA methods: An implementation on manufacturing sector, Celal Bayar University Journal of Social Sciences 11(2) (2013), 274–273.

70.

Hinduja

and Pandey

, An integrated intuitionistic fuzzy MCDM approach to select cloud-based ERP system for SMEs,, International Journal of Information Technology & Decision Making 18(06) (2019), 1875–1908.

71.

Wang

, Peng

and Wang

, A multi-criteria decision-making framework for risk ranking of energy performance contracting project under picture fuzzy environment, Journal of Cleaner Production 191 (2018), 105–118.

72.

Zhang

, Kou

, Yu

and Gao

, Consistency improvement for fuzzy preference relations with self-confidence: An application in two-sided matching decision making, Journal of the Operational Research Society, (2020), 1–14.

73.

Zhang

, Gao

and Yu

, Two-sided matching decision making with multi-granular hesitant fuzzy linguistic term sets and incomplete criteria weight information, Expert Systems with Applications 168 (2021), 114311.

74.

Sindhu

M.S.

, Rashid

and Kashif

, Modeling of linear programming and extended TOPSIS in decision making problem under the framework of picture fuzzy sets, PloS One 14(8) (2019), 1–13.

75.

Dahooei

J.H.

, Zavadskas

E.K.

, Vanaki

A.S.

, Firoozfar

H.R.

and Ghorabaee

M.K.

, An evaluation model of business intelligence for enterprise systems with new extension of CODAS (CODAS-IVIF), Economics and Management 21(3) (2018), 171–188.