This paper presents a new concept of trapezoidal cubic fuzzy numbers and solves the plant location selection (PLS) problem based on a new decision method with cubic fuzzy information captured through trapezoidal cubic fuzzy numbers. In the decision process, the unknown weights of the criteria are unearthed by using the Shannon entropy theory and the weights of the decision makers by integrating the Evidence theory with Bayes approximation. Based on trapezoidal cubic fuzzy numbers, we extend the classical VIKOR method to solve the MAGDM problems under cubic fuzzy environment based on the TrCFNs on proposed method.
In latest economical market of present time world decision making has completely become central source which have numerous unpredictable and non-commensurable approximation standards. Therefore most of the real-world decision-making problems can be viewed as multiple attribute decision-making (MADM) [2, 9] problems that are used in cases of discrete and limited number of alternatives characterized by multiple and conflicting attributes.
Furthermore in practical group decision-making, due to the complication and subjectivity of the decision systems and the undetermined source of human judgment, the evaluation results given by the decision experts are not essentially crisp numbers, but may be linguistic terms or labels of fuzzy sets [33] as for the qualitative attributes the quantification of the tentative information is actually a difficult task.
The intuitionistic fuzzy set [1] is more suitable and flexible in dealing with fuzziness and vagueness set up from fuzzy knowledge or information that involves hesitation. Shen et al. [18] proposed an outranking sorting method for group decision making using IFSs. Wan et al. [27] developed a new method for solving MAGDM problems with Atanassovs interval-valued intuitionistic fuzzy values and incomplete attribute weight information. Wang [28] defined the trapezoidal intuitionistic fuzzy number, which is an extension of the triangular intuitionistic fuzzy number. TIFNs and TrIFNs extend the domain of the IFSs from the discrete set to the continuous set. Wu and Cao [29] proposed some geometric aggregation operators for aggregating TrIFNs and applied them to MAGDM problems. Zhang et al. [34] defined a grey relational projection method for MADM problems fundamental on TrIFN. Wan [25] developed a new decision method based on power average operators of TrIFNs and applied it to MAGDM problems. Li and Chen [12] nominate a new MAGDM method using consistency analysis for the trapezoidal intuitionistic fuzzy TOPSIS method. Ye [30] established the similarity measures based on the Hamming and Euclidean distances between TrIFNs to solve the MAGDM problem. Li and Chen [13] presented an extended TOPSIS method with TrIFNs and prospect theory for MAGDM problems. Shen et al. [17] presented a arithmetic aggregation operator called I-ITFOWA and refined a MAGDM process utilizing the proposed operator and the ITFWAA operator in the intuitionistic trapezoidal fuzzy settings. Dutta and Guha [5] investigated Bonferroni mean aggregation operator under intuitionistic trapezoidal fuzzy environment and proposed a weighted MADM technique. Shi [19] investigated the dynamic MADM problems under intuitionistic trapezoidal fuzzy environment and developed a new operator called dynamic intuitionistic trapezoidal fuzzy weighted geometric averaging operator. Among the many available methods, VIKOR (VlseKriterijumska Optimizacija I Kompromisno Resenje in Serbian. To obtained compromise solutions could be the base for debate in the decision process, involving preferences of the decision-maker using criteria weights [14]. Opricovic [15] defined the Fuzzy VIKOR with an application to water resources planning. Devi [6] introduction the Extension of VIKOR method in intuitionist fuzzy environment for robot selection. Wan et al. [26] extended the classical VIKOR method to MAGDM problems with fuzzy preference information captured through TIFNs.
Cubic sets introduced by Jun et al. [10], are the generalizations of fuzzy sets and intuitionistic fuzzy sets, in which there are two representations, one is used for the degree of membership and other is used for the degree of non-membership.
The rest of the paper is organized as follows. Section 2 is devoted to present some basic concepts and results relevant to the proposed research. Section 3, we define some new concepts comprising the definition, operations, crisp weighted possibility means and hamming distance of the trapezoidal cubic fuzzy numbers (TrCFNs). In Section 4, we develop a MAGDM approach based on an extended VIKOR method using trapezoidal cubic fuzzy numbers (TrCFNs). In Section 5, we discuss implementation of the solution methodology to solve the PLS problem. A discussion of the obtained results and sensitivity analysis are also included in this section. Finally, we submit our concluding remarks in Section 6.
Preliminaries
Definition 2.1. [1] Let H be a universe of discourse. The idea of fuzzy set was presented by Zadeh and defined as follows; . A fuzzy set in a set H is defined ΓJ : H → I, is a membership function, denoted the degree of membership of the element to the set H, where I = [0, 1]. The collection of all fuzzy subsets of H is denoted by IH. Define a relation on IH as follows:
Definition 2.2. [10] Let H be a nonempty set. By a cubic set in H we mean a structure F = {h, α (h) , β (h) : h ∈ H} in which α is an IVF set in H and β is a fuzzy set in H A cubic set F = {h, α (h) , β (h) : h ∈ H} is simply denoted by F = 〈α, β〉. The collection of all cubic sets in H is denoted by CH.
Definition 2.3. [10] Let H be a non-empty set. A cubic set F = (C, λ) in H is said to be an internal cubic set if C- (h) ⩽ λ (h) ⩽ C+ (h) for all h ∈ H.
Definition 2.4. [10] Let H be a non-empty set. A cubic set F = (C, λ) in H is said to be an external cubic set if λ (h) ∉ (C- (h) , C+ (h)) for all h ∈ H.
The definition and arithmetical operations of TrCFNs
In this section, we define some new concepts comprising the definition, operations,
and hamming distance of the trapezoidal cubic fuzzy numbers, whichever are used in the subsections.
Definition 3.1. Let be the trapezoidal cubic fuzzy number on the set of real numbers, its interval value trapezoidal fuzzy number is defined as:
and its trapezoidal fuzzy number is
Then the TrCFN basically denoted by . Further, the TrCFN reduced to a TCFN. Moreover, if ωb- = 1, ωb+ = 1 and , if the TrCFN is called a normal TCFN denoted as . Then is called trapezoidal cubic fuzzy number.
Definition 3.2. Let , , and , , , be two TrCFNs and ξ be any real number. The operational rules over TrCFNs are stipulated as under:
, (u1,
-u2)]; ,
, (t1, t2) , (u1, u2)]; , , , ; ,
; , ,
; (1-
The crisp weighted possibility means of TrCFNs
Definition 3.1.1. For a TrCFNs ; , the (α, β) –cut set, the α - cut set and the β - cut set are define as , (h) ⩾ α} and
Definition 3.1.2. Let , {Lb+ (α) , Rb+ (α)}] 〉 be the a-, a+ -cut set of a TrCFN . The φ weighted lower and upper possibility means of the membership function λ-, λ+ for the TrCFN are defined as
respectively, where φ : [0, ωb-] → R- and φ : [0,ωb+] → R+ are a non-negative, monotone increasing weighted function satisfying and φ (0) =0, and φ (0) =0. Here, Pos means possibility which are defined as
using Equations (7), (8) and (9), (10), we get
Evidently, and shows the φ weighted lower possibility weighted average of the minimum of the α -cut set and hence called the φ weighted lower possibility mean of interval values. and shows the weighted upper possibility weighted average of the maximum of the α -cut set and thus called the φ weighted upper possibility medium of interval values.
Definition 3.1.3. be the β -cut set of a TrCFN . The φ weighted lower and upper possibility means of the non-membership function Γ for the TrCFN are defined as
and
respectively, where is a non-negative, monotone decreasing weighting function satisfying , and
Most likely, reflects the ψ weighted lower possibility weighted average of the minimum of the β -cut set, hence the name ψ weighted down possibility mean of non-membership function. Likewise, reflects the ψ weighted upper possibility weighted average of the maximum of the β -cut set and is called ψ weighted upper possibility mean of non-membership function.
Definition 3.1.4. For a TrCFN ; , the φ weighted possibility mean of interval value trapezoidal fuzzy number λ is defined as
and the ψ weighted possibility mean of trapezoidal fuzzy number Γ is defined as
Example 3.1.5. If we selected and respectively of the following shape
and then according to Equations (3), (4), (5) and (6), we have and according to Equations (11) and (12) we have
Further, from Equations (15), (16) and (17), it follows
and
Thus, for a TrCFN the weighted possibility mean interval can be create from and as follows:
Weighted average operator of TrCFNs
Definition 3.2.1. For TrCFNs , , , the weighted average operator TrCFN-Wa: φn → φ is defined as TrCF-Wa , , ..., , , where the set of all TrCFNs is denoted by φ, is the weight vector of and , The TrCFN-WA operator is called the trapezoidal cubic fuzzy weighted average operator.
Theorem 3.2.2.The aggregated value of TrCFNsunder the TrCFN-Wa operator is also a TrCFN given by
Proof Let be the TrCFNs.
Now, we use mathematical induction for n to prove the theorem. For n = 1, using Definition 3,i.e.,TrCFN-Wa
Now, we prove it for n = 2. Using, the case of n = 1, we have w1b1 + w2b2 as TrCFNs. Now, we obtain the aggregation of these numbers as
which is also a TrCFN using Definition 3. We now assume that it is true for n = k, i.e.,TrCFN-Wa
is a TrCFN. We prove it for n = k + 1, we have TrCFN-Wa
which is also a TrCFN using Equations (24) and (25).
Hamming distance for TrCFNs
Definition 3.3.1. Consider the Banach space X, the hausdorff distance between any two subsets E and F can be defined as . If X = R and E and F are closed intervals and , respectively, then the above distance becomes to .
Definition 3.3.2. Let b1 = [r1, s1, t1, u1]; , , ηb1〉 and b2 = [r2, s2, t2, u2]; , be two TrCFNs. The hammimg distance between and is defined as follows:
A MAGDM approach based on an extended VIKOR method using TrCFNs
Here, we first present the characterization of MAGDM problems with TrCFNs and determine the weights of decision-makers as well as attributes. A MAGDM approach based on an extended VIKOR method for TrCFNs is then proposed.
MAGDM problems using TrCFNs
Assume that there exists l non-inferior alternatives {B1, B2, . . . . Bl} to be evaluated against n attributes {b1, b2,. . . . bn} with the associated weighting vector W = (w1, w2,. . . . wn) T, satisfying wj ∈ [0, 1] (j = 1, 2,. . . , n) and . Further, suppose there exist q decision-makers {DM1, DM2,. . . . . DMn} in the experts group with the associated weighing vector as ρ = (ρ1, ρ2,. . . . ρq) T satisfying ρ ∈ [0, 1] (k = 1, 2,. . . q) and . Assume that each decision-maker provides the characteristics of the alternatives Bj with respect to each attribute bj using TrCFNs as .
Next, we determine the weights W = (w1, w2,. . . . wn) T of the attributes using Shannon entropy theory and the weights ρ = (ρ1, ρ2,. . . . ρq) T of the decision-makers using the Evidence theory with Bayes approximation.
Weights of the attributes
To determine the attribute weights, which are unknown using Equation (23), the fuzzy assessment data provided by the decision-maker for each alternative with respect to each attribute is defuzzified into the crisp weighted possibility mean values (k = 1, 2,. . . . q), where
In general, attributes classified into two types: benefit attributes and cost attributes, i.e., the set F of attributes can be divided into two sets Fc and Fd, where Fc denotes the set of benefit attributes and Fd denotes the set of cost attributes, Fc ∩ Fd = φ and Fc ∪ Fd = F.
where and . Since, by using Shannon entropy measure, we objectively determine the weights of the attributes, the crisp weighted possibilities mean values Vk need to be normalized unless all the attributes bj (j = 1, 2,. . . , n) are of the same type. we have
The entropy measure of the attribute bj with respect DMk is computed as follows:
Thus corresponding to attribute bj, the individual weights with respect DMk can be obtained as follows:
Note that to have the composite attribute weight vector W = (ω1, ω2,. . . . . , ωn) T, we need the weight vector ρ = (ρ1, ρ2,. . . ρq) T of the decision-makers.
Weights of decision-makers
Let there be a fixed set of l mutually exclusive and exhaustive propositions called the frame of discernment which is indicated by Ω = {B1, B2,. . . Bl}. A mass function over Ω (also known as basic probability assignment (BPA)) is a function M (.) : 2Ω → [0, 1] such that M (φ) =0 and ∑a⊆2ΩM (b) =1 where 2Ω represents the power set of Ω and M (b) corresponds to the measure of belief that is committed exactly to the proposition B. Voorbraak [24] defined Bayes approximation of belief functions, which is computational less involving as it distributes mass M to a subset of Ω over its elements rather than assigning the mass M on the power set of Ω as in the case of Dempster-Shafer theory thus making the amount of computation required for the combination of evidence as exponentially increasing. The Bayesian approximation M of belief function convinced by the BPA M is computed as
Where | · | denotes the cardinality function. Considering the alternatives {b1, b2,. . . . bn} as the propositions of the frame of discernment of the MAGDM problems, let be the evidence body of the proposition bi on attribute bj provided by DMk. It follows
where the assessment of alternative bi on attribute bj, are considered as evidence bodies. The weighted attribute evidence body of alternative bi on attribute bj by DMk is computed as
Incorporating combinational rule with Bayesian approximation, then n Bayes approximations (j = 1, 2,. . . n) corresponding to the mass function could be aggregated into the respective composite evidence body of alternative bi with respect to DMk as follows:
where Yi is a singleton and ∑∩Y≠φ, represents the extent of fracas between bodies of evidence. Let the bodies of evidence . The distance between any two bodies of evidence and denoted as dis can be defined as follows:
which is formulated into the similarity matrix . Obviously, similarity would be enhanced if the distance reduces. Let the support degree, denoted by , the function of similarity which represents the degree to which an evidence body is supported by other bodies of evidence, is expressed as follows:
The credibility degree, denoted by of the body of evidence represents the relative importance of the evidence given by DMk and hence measures the weight of DMk, defined as
Of course . Thus, the weight vector of DMs, ρ = (ρ1, ρ2,. . . . ρq) T is obtained as follows:
The extended VIKOR method using TrCFNs
Here, we extend the typical VIKOR method created on the beyond analysis. The steps are defined as below:
Step 1: Calculate the crisp weighted possibility mean values of decision-makers byEquation (27) from the TrCFN assessment date given by each DMk based on attribute with respect to alternative Bi.
Step 2: Forecast the individual weights for each attribute bj with respect to each DMk using Equations (28)–(31).
Step 3: Using Equations (33)–(41), determine the weights of each decision-maker ρ = (ρ1, ρ2,. . . . ρq) T.
Step 4: Compute the Combined weight vector of attribute W = (ω1, ω2,. . . ωn) T using Equation (32)
Step 5: Using the TrCFN-WA operator (i.e., Equation (23))
compute the TrCFN group decision values by aggregating the fuzzy assessment data given by each
where
Step 6: With a view to except the effect of diverse physical dimensions on the final decision values, the group decision values must b normalized into , where values, the group decision values, where
and
Also for the benefit attributes, we have
and
Step 7: The positive ideal solution and negative ideal solution, are to be resolved, where indicates an aspiration level and represents least tolerable level and are given as follows:
Step 8: Using hamming distance, compute the group utility value UG (Bi) and individual regret value RI (Bi) with respect to alternative Bi, i = 1, 2,. . . . , l as follows:
where UG represents the distance of the ith alternative to the positive ideal solution, which prove the best combination and RI represents the distance of the ith alternative to the negative ideal solution, which indicates the worst combination.
Step 9: For each alternative Bi compute the degree of closeness C (Bi) to the ideal solution as below:
Step 10: Rank the alternative in increasing order C (Bi) , (i = 1, 2,. . , l).
An application to the plant location selection problem
To highlight the utility of the extended VIKOR method proposed in the previous section, here we consider its application to the PLS problem. In the real-world PLS, it is often difficult for the decision-makers to express precisely the importance of the attributes and the impact of alternatives on attributes.
Many authors have developed precision-based MADM approaches to treat the PLS problem. Spohrer and Kmak [20] developed weight factor analysis method for the PLS problem that integrates both the quantitative data and qualitative ratings. Yoon and Hwang [32] used different MADM methods for different versions of manufacturing plant site selection problem. Tavakkoli-Moghaddam [22] proposed an integrated methodology that includes AHP and VIKOR to solve the PLS problem. Pavi'c and Babic [16] used the PROMETHEE method for the location choice of a production system.
Using fuzzy set theory concepts, many authors have developed fuzzy-MCDM methods for the PLS problem. Ertugrul and Karakasoglu [7] used the fuzzy AHP and fuzzy TOPSIS methods for the selection of facility location. Kaboli [11] presented a holistic MCDM approach to select the optimal location(s) that best-fit for both the investors and managers. Yong [31] proposed a TOPSIS approach for the PLS problem under linguistic environment. Chou et al. [4] developed a fuzzy simple additive weighting system to solve the facility location selection problem using objective/subjective attributes under group decision-making conditions. Chaudhary and Shankar [3] proposed an STEEP-fuzzy AHP-TOPSIS based framework for evaluation and selection of optimal locations for the thermal power plants.
To better address highly uncertain problems such as the PLS problem, we need to incorporate the uncertainty during the alternative’s evaluation and also ranking the alternatives. Since the trapezoidal cubic fuzzy numbers are special cubic fuzzy sets defined on a real number set and are considered useful to deal with vagueness and imprecision in the decision data and decision-making problems, we use trapezoidal cubic fuzzy numbers to obtain information from the decision-makers in respect of various parameters related to the PLS problem and solve it by the proposed extended VIKOR method.
Assume that a company is looking to select a location to build a new plant. The geographical location of the plant can have a strong influence on the success of the industrial project. Considerable care must be exercised in selecting the plant site and many different factors must be considered. Primarily the plant must be located where the minimum cost of production and maximum distribution can be obtained but, other factors such as room for expansion and safe living conditions for plant operation as well as the surrounding community are also important.
There are various attributes that must be considered while selecting a suitable location for a plant, but here, we have considered the following six attributes studied.
Expert labors (a1)
Development probability (a2)
Shoot length (a3)
Fresh weight (a4)
Natives
The experts of the government evaluate and over their own opinions regarding the results obtained with each alternative. The weights of experts of is given as w = (0.4, 0.2, 0.2, 0.2) As the environment is very uncertain, the group of experts needs to assess the available information by using TrCFNs. The expected results given in the form of TrCFNs depending on the characteristic Cj and the alternative Ai are shown Table 1. Such that,
Attributes
DM1
C1
C2
C3
C4
A1
EG
VVG
MH
EH
A2
MG
F
EG
EH
A3
MH
MG
MG
EH
A4
VVG
EH
MH
M
Step 1: We take BUM function p (t) = t (s = 1): Based on the fuzzy assessment data provided in Table 2, we obtain the crisp weighted possibility mean values of decision-makers using Equation (27) and these are listed in DM1
Attributes
DM1
C1
C2
C3
C4
A1
A2
A3
A4
Attributes
DM1
C1
C2
C3
C4
A1
A2
A3
A4
Attributes
DM1
A1
0.4480
0.2328
A2
0.4233
0.4868
A3
0.3747
0.2481
A4
0.3832
0.2442
The normalized possibility mean values
Step 2: The normalized possibility mean decision values of decision-makers are recall as in DM1 using Equations (30) and (31). Further, using Equations (32) and (33), we obtain the entropy values and weights for each attribute with respect each decision-maker and are provide in DM1. The entropy values and weights for each attribute
Step 3:Now, we obtain the weighted attribute evidence bodies aj given by DMk using Equations (35) and (36) and are provided in DM1. The weighted attribute evidence bodies
Now, the Bayes approximation of mass function M. are obtained using Equation (33) and are provided in DM1. The Bayes approximation of mass function
Using Equation (37), the Bayes approximation of alternative Bi are aggregated into the individual overall evidence body for and are obtained as follows
Using Equations (38) and (39), the evidence distance matrix between and (u, v = 1, 2, 3, 4) is computed and using Equation (40), the evidence similarity matrix is obtained as follows: D = [0.5672, 0.6051, 0.9288, 0.6697]
using Equation (41), the supports degree of is calculated as follows:
. Thus, using Equations (41) and (42), we obtain the weight vector of decision-makers as follows:
Step 4: Using Equation (32), the weighted vector of attribute is now obtained as follows
Step 5: Using Equation (43), the TrCFN-WA group decision values (H) are provided in DM1. The DM1 TrCFN-WA group decision values
Step 6: Since the attribute b4 is a cost attribute and rest all are benefit attributes, we obtain the normalized group decision values X using Equations (44) and (47) provided in DM1:
Step 7: Now, we obtain the positive ideal solution and negative ideal solution of the normalized group decision values using Equations (48) and (49), respectively. PISNIS
Step 8: Using the hamming distance Equation (26), compute the group utility values and individual regret values Equations (50) and (51) as follows:
Based on the above values, we have
Step 9: Using Equation (53), we calculate the closeness of alternative Bi to the ideal solution provided in Table 14
Step 10: The ranking order of alternatives with coefficient of decision mechanism τ = 0.5 in increasing order of C (Bi) , i = 1, 2, 3 is listed in DM1. Ranking of alternatives
Conclusion
Based on the fact that most of the real-world decision situations involve evaluation of the alternatives on multiple conflicting attributes using vague and imprecise assessments provided by a group of decision-makers, we proposed a new decision method to solve the fuzzy MAGDM problems. We assumed that the ratings of alternatives on the given attributes are expressed using TrCFNs and the weights of attributes and decision-makers’ are completely unknown. Note that TrCFNs are extensions of fuzzy numbers and CFSs, which can express more abundant and flexible information. The weights of attributes are calculated using Shannon entropy measure and the weights of decision-makers’ are determined by integrating the evidence theory with Bayes approximation. The group decision matrix is obtained by an appropriate aggregated process that combines the information provided by all the decision-makers. Since VIKOR method is considered a useful technique to treat the MAGDM problems that have non-commensurable and conflicting attributes based on providing a maximum group utility for the majority and a minimum individual regret for the opponent, we extended the classical VIKOR method to treat fuzzy MAGDM problems under cubic fuzzy environment involving TrCFNs. The utility of the proposed method was demonstrated by solving the real-world MAGDM
Problem, namely the PLS problem under the above stated assumptions. The numerical illustrations based on real-world data clearly established advantages of the proposed extended VIKOR method over many similar methods such as ELECTRE method. It is worth mentioning that besides the PLS problem, the proposed methodology is shown appropriate and advantageous for MAGDM in many other important areas, such as supplier selection and software selection.
Considering the importance of TrCFNs in real-world MAGDM problems, the proposed decision method can be applied to solve many other practical MAGDM problems arising in different areas of management science.
Footnotes
Acknowledgments
The author M. Aslam would like to express his gratitude to deanship of scientific research at King Khalid University, 61413, Abha, Saudi Arabia for providing administrative and technical support.
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