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Healthcare evaluation in hazardous waste recycling using novel interval-valued intuitionistic fuzzy information based on complex proportional assessment method
Journal Pre-proofs Healthcare Evaluation in Hazardous Waste Recycling Using Novel IntervalValued Intuitionistic Fuzzy Information Based on Complex Proportional Assessment Method Arunodaya Raj Mishra, Pratibha Rani, Abbas Mardani, Kamal Raj Pardasani, Kannan Govindan, Melfi Alrasheedi PII: DOI: Reference:
S0360-8352(19)30609-6 https://doi.org/10.1016/j.cie.2019.106140 CAIE 106140
To appear in:
Computers & Industrial Engineering
Please cite this article as: Raj Mishra, A., Rani, P., Mardani, A., Raj Pardasani, K., Govindan, K., Alrasheedi, M., Healthcare Evaluation in Hazardous Waste Recycling Using Novel Interval-Valued Intuitionistic Fuzzy Information Based on Complex Proportional Assessment Method, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/j.cie.2019.106140
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Healthcare Evaluation in Hazardous Waste Recycling Using Novel Interval-Valued Intuitionistic Fuzzy Information Based on Complex Proportional Assessment Method Arunodaya Raj Mishra1, Pratibha Rani2, Abbas Mardani3, Kamal Raj Pardasani4, Kannan Govindan5, Melfi Alrasheedi6 1Department
2Department
3Department
4 Department
5
of Mathematics, Govt College Jaitwara, Satna-485221, M P, India, email: [email protected]
of Mathematics, National Institute of Technology, Warangal, Telangana-506004, India, email:[email protected]
of Marketing, Muma Business of School, University of South Florida, Tampa, FL 33813, United States, email: [email protected]
of Mathematics, Bioinformatics and Computer Applications, MANIT, Bhopal-462051, M P, India, email: [email protected]
Center for Engineering Operations Management, Department of Technology and Innovation, University of Southern Denmark, Denmark DK-5230 Odense M, Denmark, email: [email protected]
6 Department
of Quantitative Methods, School of Business, King Faisal University, Saudi Arabia, email: [email protected]
Abstract Nowadays, the selection of the appropriate safety and health evaluation facility in hazardous waste recycling organizations has become a challenging issue, mainly in developing countries. Evaluation of hazardous waste recycling facility selection can be considered as a complex multicriteria decision-making problem that contains multiple-options solutions with conflicting tangible and intangible criteria. This paper is proposed a novel method based on the Complex Proportional Assessment (COPRAS) with multi-criteria decision making for interval-valued intuitionistic fuzzy sets. The proposed method is derived from aggregation operators of interval-valued intuitionistic fuzzy sets; in addition, a few amendments in the conventional complex proportional assessment approach and a process for computation of criteria weights are presented. For calculating the weights of the criteria the interval-valued intuitionistic fuzzy information measures are proposed. The present approach mainly addresses the correlative multi-criteria decision-making problems; therefore, the criteria weights are calculated in the form of Shapley values. Recently, increasing challenges for environmental issues require the development of hazardous waste recycling facility selection with respect to different criteria; therefore, the practical problem of hazardous waste recycling facility selection is presented, which proves the validity and feasibility of the proposed approach. Finally, a comparison is made with existing approaches to validate the developed method. Keywords: Information measures; Safety and health; Hazardous waste; Interval-valued intuitionistic fuzzy sets; Complex proportional assessment; Multi-criteria decision making.
1. Introduction A variety of wastes and sub-products are generated by various industrial sectors, such as food, healthcare, and mining (Guimarães et al., 2018). Roughly 40% of the wastes generated by these sources are classified as hazardous, which must be well treated in a way to prevent from appearing negative impacts on the environment and people’s health (Guimarães et al., 2018). In many industries like healthcare, it is important to avoid hazards, and a healthy environment is necessary for delivering safe outcomes. As a result, risk assessment is an important activity for many organizations through which they attempt to explore the methods for the mitigation or avoidance of hazards and risks that may arise within their site (Nahrgang et al., 2011). Remember that the majority of biological hazards come fromthe facilities used for waste treatments or the place wherein people are handling manually the untreated waste. During the waste treatment process, workers might be in direct or indirect contact with body fluids and blood that may leak out of used containers. In this procedure, they may also be in contact with airborne pathogens. The assessment of health and safety conditions in the work place with risky facilities of waste recycling is a complex MCDM problem that involves multiple and often contradictory qualitative and quantitative criteria (Guo et al., 2018; Hu et al., 2019; Tsai C. Kuo, 2013; Tremblay & Badri, 2018). Cost and benefit analysis is the general model for the analysis of social and private decisions. The cost and benefit analysis identify, measure and aggregate the costs and benefits to rank and select the optimal alternatives (Dompere, 1995). There are different subjectivity, vagueness and imprecision characteristics that should be accounted in different perspectives in social costing (Rothenberg, 1969). The concept of cost-benefit analysis is typically used to assess the environmental safety and health evaluation (Abrahamsen et al., 2018; Riaño-Casallas & Tompa, 2018). In addition, an economic index like the cost-benefit ratio is employed to compare various available decision alternatives. Such an index can also show uncertainty through the incorporation of the expected cost offailure. The extensive use of the risk-cost-benefit formula is due to its straightforwardness and simple interpretation of the obtained outcomes in monetary terms. Khadam and Kaluarachehi (2004) discussed in detail the limitations of the risk-cost-benefit approaches that are currently applied to assessing the environmental safety and health status. In the theory of fuzzy sets (FSs), intuitionistic fuzzy sets (IFSs), and interval-valued intuitionistic fuzzy sets (IVIFSs), the information (divergence and entropy) measures play
aimportant role in the study of uncertainty (Arya & Yadav, 2018; Çalı & Balaman, 2019; Sirbiladze et al., 2018; Zhang et al., 2019). At first, Zadeh (1969) studied the notion of entropy measure in FSs. Confirming the probability entropy, De Luca and Termini (1972) firstly presented the axiomatic definition of fuzzy entropy. Pal and Pal (1989) defined an exponential type entropy measure for FSs. Next, Burillo and Bustince (1996) originated the idea of IF-entropy and IVIFentropy. Furthermore, Szmidt and Kacprzyk (2001) pioneered IF-entropy with their geometrical interpretation. Hung and Yang (2006) developed the IF-entropy using the axioms of probability. Afterward, different researchers would analyze the concept of entropy in FSs and IFSs (Mishra & Rani, 2019; Narayanamoorthy et al., 2019; Qi et al., 2015; Zhang et al., 2019). Divergence measure is an essential procedure to estimate the distinction between elements. In FSs philosophy, Bhandari and Pal (1993) pioneered the notion of divergence measure. Similar to FSs, Vlachos and Sergiadis (2007) established the idea of IF-divergence measure and applied it to the image segmentation, medical problems, and pattern recognition. Next, Montes et al. (2015) presented a divergence measure for IFSs with new definitions and properties. Mishra et al. (2017) proposed new Jensen-exponential divergence measures; then, applied them to the development of a new MCDM technique with incomplete criteria under the IFSs environment. Mishra et al. (2019c) extended the divergence measure for hesitant fuzzy sets and utilized them to develop the WASPAS approach under hesitant fuzzy sets context. Further, Rani et al. (2019a) studied new divergence measure for Pythagorean fuzzy sets and applied to focus on the development of Pythagorean fuzzy VIKOR method. Currently, the concept of IVIF-divergence measure is extended, and until now, only a few researchers have focused onIVIF-divergence measure (Mishra, Chandel, et al., 2018; Mishra & Rani, 2018; Mishra, Singh, et al., 2019; Pratibha Rani et al., 2019). Based on compromise programming, one of the new methods called Complex Proportional Assessment (COPRAS), originated by Zavadskas et al. (1994), is a sensible and well-organized approach to processingthe information. The COPRAS approach has some key features as follow: It provides a valuable and suitable way to find a solution tothe MCDM problems (Anthony et al., 2019; Dey et al., 2017; Garg & Nancy, 2019; Roy et al., 2019; Wang et al., 2016). It assumes both aspects of criteria using Complex Proportional Evaluation, which provides moreexact information compared to different approaches proposed basically for solving the benefit or cost criteria. Finally, it describes the ratios to the optimal and the worst solutions at the same time.
Recently, due to the increasing complexity of MCDM problems, some authors extended the classical COPRAS method within the context of FSs, IVFSs, IT2FSs, HFSs and hesitant fuzzy linguistic sets (Mukhametzyanov & Pamucar, 2018). For instance, Bekar et al. (2016) introduced a new COPRAS-based technique with grey relations to evaluate the fuzzy MCDM Total Productive Maintenance model. Hajiagha et al. (2013) developed the COPRAS approach under an uncertain environment where the uncertainty is depicted in terms of IVIFSs. Keshavarz Ghorabaee et al. (2014) studied a new COPRAS-based approach to evaluate the supplier selection problem under Interval Type-2 fuzzy sets (IT2FSs). Vahdani et al. (2014) introduced the COPRAS method reflecting the subjective opinions and objective information of the real-life MCDM problems. Wang et al. (2016) proposed the COPRAS and Analytic Network Process (ANP)-based Failure Modes and Effects Analysis (FMEA) model for assessing and grading the risk of failure modes within the IVIF context. Zheng et al. (2018) extended the COPRAS approach with HLTSsto evaluate the severity of the chronic obstructive pulmonary disease. Mishra, et al. (2019) introduced an analytical COPRAS approach to explain the MCDM problems under HFSs. As the generalization of IFSs, IVIFSs are the vast growing environment of research, and these methodsare more flexible intackling the uncertainty and imprecision that may occur in daily life problems. Due to its capability, the present work focuses on the IVIF environment. Entropy and divergence measures, which are two critical topics in the FS theory and IVIFS theory, are widely implemented to different areas of decision making, medical diagnosis, and image segmentation. Thus, it is exciting to study the entropy and divergence measures of IVIFSs. As the exponential function has an advantage over the existing ones, this paper proposes the new IVIF-entropy and IVIF-divergence measures. However, a review of the literature shows that the entropy and divergence measures-based COPRAS approach has not been proposed under the IVIF environment, where the characteristics among the criteria are interdependent.Therefore, this paper proposes the IVIF-COPRAS method to handlethe problem of healthcare evaluation in hazardous waste recycling. The most significant contributions of this study are presented as follow: (i)
The novel entropy and divergence measures for IVIFSs are introduced and compared with existing ones.
(ii) Recently, various MCDM methods have been developed, in which the criteria have interrelated characteristics. However, these conjectures are despoiled in various cases. In this study, to cope with interdependent characteristics among the criteria, we developed Shapley values to evaluate the significance of criteria. (iii) The classical COPRAS method is extended with IVIFSs, in which the Shapley values are considered as the criteria weights. (iv) The developed method is applied to hazardous waste recycling facility selection and the performance of the proposed approach is compared with various existing ones. The paper is structured as follows: Section 2 provides the literature review and application decision making for evaluation of safety and healthy problems in hazardous waste recycling. Section 3 shows the fundamental concepts related to IVIFSs and information measures. Section 4 presents new IVIF-divergence and IVIF-entropy measures with counter-intuitive cases of existing measures. Section 5 proposes the COPRAS method based on Shapley function and entropy and divergence measures under the IVIF-environment. Also, a decision-making problem of HWR facility selection with IVIFSs is presented. Further, a comparision and sensitivity analysis are discussed to enlighten the applicability and feasibility of the proposed approach. Finally, Section 6 presents the conclusions of the whole study. 2. Evaluation of safety and health status in hazardous waste recycling People collect wastes and work with waste recycling systems may be subjected to a variety of risks and infections that, in many cases, may remain untreated. This may occur because of a limitation in having access to healthcare facilities. Therefore, to enhance the job-related safety of those who collect and recycle wastes is a crucial step towards the improvement of their social welfare. For this purpose, in the first step, there is a need for finding the actual job-related risks that are generally accompanied by any activity related to managing and working with solid wastes (Bleck & Wettberg, 2012). This is because of the fact that numerous toxic substances are possible to be given off in the course of recycling and disposing, which can simply lead to negative impacts on the environment, hence being a threat to people’s health status (Zheng et al., 2013). Such toxic substances are of a high risk not only to human beings but also to the whole environment and ecosystem (Cao et al., 2016; Long et al., 2013) through a variety of exposure ways (Song & Li, 2015; Zeng et al., 2016). Generally, workers are subjected to such negative impacts by being
exposed directly through their skin and inhalation system, whereas other people may be exposed through dust, smoke, food, and drinking water contamination (Robinson, 2009). Waste comes from a variety of sources, including the operations done for pollution control or decontamination processes, substances from contaminated soil remediation activities, adulterated materials, and activities done to purify the raw materials. The list can be also extended toalkaline or acidic substances, oils, sludge, plastics, wood, ashes, slag, metal, fibers, paper, glass, and rubber (Nakomcic-Smaragdakis et al., 2016). The research into this issue was carried out in a developed country, while the condition of developing countries has remained unknown, where the hazard can be more significant. Heavy metals, for instance, cadmium, chromium lead, zinc, and copper are some of the significantly-dangerous substances that can comeout of the waste recycling systems. They can be hardly removed from nature; they can be accumulated in human beings’ vital organs. The developed countries have made it a legal obligation to recover materials from waste flows in a formal way (Favot et al., 2016; Morris & Metternicht, 2016; Zhou et al., 2017). In these countries, strict regulations have been provided for waste recycling processes aiming at reducing the negative environmental impacts (Cesaro et al., 2018). On the contrary, developing countrieshave various informal recycling methods (Ardi & Leisten, 2016; Salhofer et al., 2016) such as mechanical operations, burning waste in open spaces, and chemical leaching processes, which are often carried out with no formal control. A piece of waste is hazardous in case it has any one of the following features: toxicity, reactivity, corrosiveness, or ignitability. Materials possessing such characteristics bring about the immediate or long-lasting risk to human beings, plants, animals, and the whole environment. Two factors that have a significant contribution to the stable increase of hazardous wastes are the growth of world population and the rise of industrialization. Nowadays, the government, the general public, and industrial sectors are aware of the pressing need for dealing effectively with the problem of hazardous wastes (Guo et al., 2018; Hu et al., 2019). Among the most extensively used methods for wastes disposition are disposition, incineration, and landfilling. On the other hand, burning and deposition of hazardous waste can, in turn, lead to an intense strain upon the environment, which can be because of the potential pollution of the water resources and the carbon dioxide that can be released from incineration sites. Hazardous wasteis given off into environment threats people's health through contaminated water and air. Recycling is known as one of the efficient ways that can manage hazardous wastes (Orloff & Falk, 2003). HWR facilities are capable
of treating, storing, or disposing of such wastes. If potential risks to safety and health are effectively prevented, workers will be able to work with the HWR facilities with a minimum level of risk. According to Cloquell-Ballester et al. (2007), the current decision-making approaches applied to the management of hazardous wastes generally have two disadvantages: the compensation problem and the problem of identifying an “exact boundary” between sub-ranges. They introduced an Elimination Et Choix Traduisant la REalité (ELECTRE) method to evaluate the importance of environmental effects based on comparative and sensitivity analyses. In another study, an MCDM model was employed by Rousis et al. (2008) to assess and rank 12 electrical waste management systems used in Cyprus. Their findings showed that the best system applicable to the Cyprus context is to partially disassemble and send the recyclable substance to the native market and disposethe residues at landfill spots. A multi-criteria mixed-integer linear programming model was introduced by Erkut et al. (2008) as a solution to the location-allocation problem of managing solid waste at a regional level. They made a comparison between the regional and prefectural planning for managing the solid wastes in Central Macedonia. On the basis of fuzzy logic, Peche and Rodríguez (2009) suggested an approach for the assessment of the environmental impacts coming from performing the projects and activities in cases wherethere is only limited information concerning the impacts. They applied fuzzy numbers to the environmental impact evaluation model to describe the impact properties. In addition, the MCDM technique was used by Dursun et al. (2011a) in order to perform an analysis based on the multi-level hierarchical structure and fuzzy logic in a way to effectively evaluate the alternatives to healthcare waste treatment. Taking into consideration the basis of fuzzy logic and multi-level hierarchy structure, Dursun et al. (2011b) attempted to evaluate the technologies of waste disposal. In addition, Liu et al. (2013) initiated a novel method called VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) to select the healthcare wastes (HCW) treatment. They validated their approach by applying it to solving the HCW selection problem in China. In another study, Liu et al. (2014) also used interval 2-tuple linguistic numbers (I2TLNs) using Optimization Method by Ratio Analysis (MOORA) plus full multiplicative form (MULTIMOORA) (Brauers & Zavadskas, 2010) method (ITL-MULTIMOORA) in a way to choose a proper technology for waste treatment in Shanghai, China. In addition, Voudrias (2016) provided wide-ranging details regarding the technologies of HCW treatment and also assessed
them employing analytic hierarchy process (AHP). Lee et al. (2016) carried out a study in England wherein AHP was employed as an instrument for exploration of the “optimal” technique that can be adopted for HCW treatment. Shi et al. (2017) introduced a hybrid MCDM method originating from the Multi-Attributive Border Approximation Area Comparison (MABAC) approach and used I2TLNs for the purpose of assessing and choosing the “best” method for HCW disposal. They also took into consideration the perception of numerous stakeholders. Xiao (2018), who introduced a D number theory-based approach, conducted one of the most recent investigations on the HCW treatment technologies. More recently, Liu et al. (2019) proposed an approach using Hamy-mean operators and intuitionistic uncertain linguistic variables for the evaluation of the best equipment of HCW treatment. Hinduja and Pandey (2018) developed a hybrid framework using the decisionmaking trial and evaluation laboratory (DEMATEL), ANP, and AHP for IFSs to predict and selectthe best HCW technique for Chhattisgarh, India. Hatami-Marbini et al. (2013) proposed anMCDM model, known as fuzzy-AHP-ELECTRE, based on an extended fuzzy method in the context of HWR. 3. Basic Concepts 3.1 Preliminaries As the FSs initiated by (Zadeh, 1965) are more capable to handle the uncertainty, these FSs have received extensive attention from scholars in the decision-making field of study. (Atanassov, 1986)) pioneered the notion of IFSs, which are distinguished by three parameters: belongingness degree, non-belongingness degree, and hesitation margins. As a result of ever-increasing intricacy of socio-economic surroundings, various researchers have extensively applied IFSs to MCDM problems (Mishra et al. 2017b; Stanujkic and Karabasevic 2018, Mishra and Rani 2019). In several cases, it is very complex to evaluate the exact degrees of these functions; thus, the degrees may be considered as intervals. Based on this idea, Atanassov and Gargov (1989) developed the notion of IVIFSs, which are expressed by intervals rather than the exact numbers and depicted by the interval belongingness, non-belongingness, and hesitation functions. Compared to IFSs, the IVIFSs are more prominent, dealing with uncertainty and imprecision because of their capability to handle the hesitant information. As a result, the utilization of IVIFSs has received more attention from several authors.
Recently, many researchers have focused their attention on the development of novel MCDM approaches under the IVIF context. To measure the uncertainty degree in IVIFSs, Liu et al. (2005) introduced the axiomatic conditions that an IVIF-entropy should comply with. Chen et al. (2010) developed an IVIF-entropy measure with its axiomatic properties and application in MCDM. Wei and Zhang (2015) defined IVIF-entropy and applied it to finding the criteria’s weights in MCDM problems. Rani et al. (2018) introduced IVIF-entropy and IVIF-similarity measures with their application in MCDM. Mishra, et al. (2018) introduced the bi-parametric IVIF-entropy and IVIFsimilarity measures and then utilized them to develop a Portuguese acronym for Interactive MultiCriteria Decision Making (TODIM) technique under IVIF context. For instance, Zhang et al. (2010) studied an IVIF-divergence measure and then utilized it to solve the pattern recognition problems. Ye (2011) developed a fresh MCDM method with IVIF-divergence measures. In another study, Meng and Tang (2013) presented an IVIF-divergence measure, which is used to assess the criteria and experts’ weights in the interactive MCDM method. (Meng & Chen, 2015) developed an MCDM approach based on novel IVIF-divergence measure and Shapley mapping associated with 2-additive measures. In addition, an IVIF-entropy and IVIF-divergence measures were developed by Mishra and Rani (2018), and they were used to discuss the application of weighted aggregated sum product assessment (WASPAS) approach. In MCDM approaches, evaluation of decision experts (DEs),weight vectors, and aggregation of information for options are the significant aspects for DEs. Literature also consists of many approaches proposed to evaluate the criteria weights and aggregation of the information under different fuzzy atmospheres. To compute the criteria weights, the past studies mentioned that diverse methods are based on the presumption that all the members of the set are independent and criteria have an equal significance. Thus, to handle the correlative MCDM problems, Sugeno (1974) originated the fuzzy measure that is a prominent device to model the interaction features among the members. For correlating the MCDM problem, the Shapley values are employed to quantify the importance of each criterion in all coalitions with identical probability (Shapley, 1953). In this study, an MCDM method is proposed under IVIFSs with interdependent interactions among the criteria. Here, various notions associated with IVIFSs and information measures are discussed.
Definition 3.1 (Atanassov and Gargov (1989): An IVIFS S on discourse set U u1 , u2 ,..., u p
is defined by
S ui , mS (ui ), nS (ui ) : ui U ,
(1)
where mS , nS :U 0,1 such that sup(mS (ui )) sup(nS (ui )) 1. The intervals mS (ui ) and
nS (ui ) represent the belongingness and non-belongingness degrees of an object ui in U , respectively. To make it easier, consider mS (ui ) mS (ui ), mS (ui ) and nS (ui ) nS (ui ), nS (ui ) , then
(2)
S ui , mS (ui ), mS (ui ) , nS (ui ), nS (ui ) : ui U , where 0 m S ( u i ) n S ( u i ) 1, m S ( u i ) 0
nS (ui ) 0.
and
The
interval
S (ui ) S (ui ), S (ui ) 1 ms (ui ) nS (ui ), 1 mS (ui ) nS (ui ) represents the hesitancy degree of u i to S . For a given ui U , the pair mS (ui ), nS (ui ) is said to be IVIFN (Xu, 2007). For simplicity, an IVIFN is expressed as S , , , , where , 0,1 , , 0,1 and 1. Definition 3.2 (Atanassov and Gargov (1989): Let S , T IVIFSs U , then different operations on IVIFNs are given by (a) S T iff mS (ui ) mT ( vi ), mS (ui ) mT (ui ), nS (ui ) nT (ui ) and nS (ui ) nT (ui ) for each u i U ;
(b) S T iff S T and S T ;
c (c) S ui , nS (ui ), nS (ui ) , mS (ui ), mS (ui ) : ui U ;
ui , mS (ui ) mT (ui ), mS (ui ) mT (ui ) , (d) S T : ui U ; nS (ui ) nT (ui ), nS (ui ) nT (ui )
ui , mS (ui ) mT (ui ), mS (ui ) mT (ui ) , (e) S T : ui U . nS (ui ) nT (ui ), nS (ui ) nT (ui )
Definition 3.3 (Liu et al., 2005): A mapping H : IVIFS U 0,1 is an IVIF-entropy if it holds the axioms: (EN1). H ( S ) 0 iff S is a crisp set; (EN2). H ( S ) 1 iff mS (ui ), mS (ui ) nS (ui ), nS (ui ) , for all ui U ;
(EN3). H S H S c ; (EN4). H ( S ) H (T ) if S T when mT (ui ) nT (ui ) and mT (ui ) nT (ui ) for each ui U or T S when mT (ui ) nT (ui ) and mT (ui ) nT (ui ) for each ui U . Definition 3.4 (Montes et al. (2015): A real function I : IVIFSs U IVIFSs U is called IVIF-divergence measure if it holds the given axioms: (D1). I S , T I T , S ; (D2). I S , T 0 iff S T ; (D3). I S W , T W I S , T , W IVIFSs U ; (D4). I S W , T W I S , T , W IVIFSs U . Definition 3.5 (Xu, 2007): Let S , , , be an IVIFN and 0, then S 1 1 , 1 1 , , .
(3)
From Eq.(1), the definition suggested byZ. Xu (2007)is employed as follows: Consider a set S S1 , S 2 ,..., S of ' ' IVIFNs, where S k k , k , k , k , k 11 . Then,IVIF-weighted arithmetic operator is given by k k k k k Sk 1 1 k , 1 1 k , k , k . k 1 k 1 k 1 k 1 k 1
Definition 3.6 (Xu, 2007): Consider S , , , be IVIFN. Then
(4)
S
(5)
1 1 , S , 2 2
are the score and the accuracy degrees of S , respectively, such that S 1,1 and S 0,1 .
Next, Xu et al. (2015) analyzed the score and accuracy functions and developed a modified score function for IVIFNs as follows: Definition 3.7 (Xu et al., 2015): Let S , , , be IVIFN. Then
* S
(6)
1 S 1 , S 1 S , 2
are said to be the normalized score and uncertainty values, respectively. Evidently, * S 0,1 and S 0, 1 .
Let S1 1 , 1 , 1 , 1
and S 2 2 , 2 , 2 , 2
be two IVIFNs. Then, a system is
described the comparability between two IVIFNs according to * S and S as follows: If * S1 * S 2 , then S1 S 2 , If * S1 * S 2 , then (i) if S1 S 2 , then S1 S 2 ; (ii) if S1 S 2 , then S1 S2 . In recent decades, various studies are available in theliterature indicating the postulation thatthe criteria have equal importance, the features under the criteria are independent, and the criteria weights are obtained by utilizing additive measures. However, these postulations have failed to deal with interrelated features among the criteria. To establish the interactions under interrelated criteria, Sugeno (1974) developed the idea of fuzzy measure (f-measure) as follows: Definition 3.8 (Sugeno, 1974): A map : P U [0,1] is said to beaf-measure on U if it holds the axioms:
(i) 0, U 1; (ii) If G T , then G T , G, T U . In MCDM problems, the term G is represented as the importance of criterion set G. The measure becomes an additive measure when G j , for any G P U . j G
In the correlated MCDM problems, the importance of each criterion is not only decided by itselfbut also gets the influence ofother criteria. To describe the overall interaction amongst the criteria, the Shapley function has been studied by many researchers. Definition 3.9 (Shapley, 1953): The Shapley function is an essential interaction index, which is defined by
j ,U
K U \ u j
n k 1! k ! n!
K u j K , u j U ,
(7)
where is a f-measure and K is distinguishedas a coalition shaped by the players in game theory, and it is observedas an association of criteria in the decision-making process. The Shapley value j is construedas a set of average importance of the concern of an object u j in any coalitions K \ u j . Furthermore, if there is no interaction, then
j ,U u j .
Property 3.1: If : P(U ) [0,1] is fuzzy measure, then j , U 0, u j U ; j 1(1) q. Property 3.2: If : P(U ) [0,1] is a fuzzy measure, then
q
,U U 1. j 1
j
Thus, j , U : j 1(1)q is a criterion weight vector.
3.2 Existing IVIF-Divergence Measures To quantify the degree of dissimilarity between IVIFSs, a variety of existing divergence measures from literature have been taken into consideration. According to Zhang et al. (2010):
p I Z JL ( S , T ) mS (ui ) ln i 1
1 mS (ui ) ln
(8)
mS (ui ) 1 2 mS (ui ) mT (ui )
1 mS (ui ) , 1 2 2 mS (ui ) mT (ui )
where mS (ui ) and mT (ui ) represent the average membership degree of ui to S and T , respectively, such that
mS (ui )
(9)
nS (ui ) nS (ui ) 1 mS (ui ) mS (ui ) 1 2 2 2 mS (ui ) mS (ui ) 2 nS (ui ) nS (ui ) , 4
nT (ui ) nT (ui ) 1 mT (ui ) mT (ui ) mT (ui ) 1 2 2 2
mT (ui ) mT (ui ) 2 nT (ui ) nT (ui ) . 4
Ye (2011) proposes Eq. (10) as follows: (10)
p m (u ) mS (ui ) 2 nS (ui ) nS (ui ) IY ( S , T ) S i 4 i 1
log 2
mS (ui ) mS (ui ) 2 nS (ui ) nS (ui )
1 2 mS (ui ) mS (ui ) 2 nS (ui ) nS (ui ) mT (ui ) mT (ui ) 2 nT (ui ) nT (ui )
p n (u ) nS (ui ) 2 mS (ui ) mS (ui ) S i 4 i 1
log 2
. 1 2 nS (ui ) nS (ui ) 2 mS (ui ) mS (ui ) nT (ui ) nT (ui ) 2 mT (ui ) mT (ui ) nS (ui ) nS (ui ) 2 mS (ui ) mS (ui )
Meng et al. (2013) suggests the following:
1 m (2um)S(nui )(u ) mS (ui ) 1 p S i S i log 2 I MT ( S , T ) mS ( ui ) p i 1 mS (ui ) nS (ui ) 1 m (u ) n (u ) m (umT) (uni ) (u ) S i S i T i T i
(11)
S
n (ui ) log 2 mS (ui ) nS (ui ) 1
, nS ( ui ) nT ( ui ) mS ( ui ) nS ( ui ) mT ( ui ) nT ( ui ) 1 m (2unS) (uni )(u ) S
i
i
S
where mP (ui ) mP (ui ) mP (ui ) and nP (ui ) nP (ui ) nP (ui ) for P S , T . And Meng et al. (2015) proposes Eq. (12) as follows: (12)
m (u ) 1 mS (ui ) I MC ( S , T ) S i log 2 2 1 12 mS (ui ) mT (ui ) i 1 p
2 mS (ui ) 3 mS (ui ) log 2 2 1 12 4 mS (ui ) mT (ui )
3 mS (ui ) nS (ui ) 2 mS (ui ) nS (ui ) log 2 2 1 12 4 mS (ui ) mT (ui ) nS (ui ) nT (ui )
mS (ui ) nS (ui ) 3 mS (ui ) nS (ui ) log 2 2 1 12 mS (ui ) mT (ui ) nS (ui ) nT (ui )
2 nS (ui ) 3 nS (ui ) nS (ui ) 1 nS (ui ) log log 2 2 2 2 1 12 4 nS (ui ) nT (ui ) 1 12 nS (ui ) nT (ui )
,
where mP (ui ) mP (ui ) mP (ui ) and nP (ui ) nP (ui ) nP (ui ) for P S , T . Next, the given examples illustrate the limitations of the existing divergence measures: Example 3.1: Consider S , T IVIFSs U where S u i , 0.1, 0.3 , 0.1, 0.3 : u i U and T
u , 0.3, 0.4 , 0.3, 0.4 i
: u i U . Then,
we
I Z JL S , T 0, IY S , T 0 and
obtain
I MT S , T 0, i.e., the measures (8)-(11) do not satisfy the postulate (D2) of Definition 3.5. Remark 3.1: In general, let us consider
T u , m (u ), m (u ) , n (u ), n (u ) m (u ), m (u )
S ui , mS (ui ), mS (ui ) , nS (ui ), nS (ui ) mS (ui ), mS (ui ) : ui U , i
T
i
T
i
T
i
T
i
T
i
T
i
: ui U .
From Example 3.1, we obtain that, I ZJL ( S , T ) 0, IY (S , T ) 0 and I MT ( S , T ) 0, whenever S T , the measures (8)-(11) violate the postulate (D2) of Definition 3.5.
Example 3.2: Consider S , T IVIFSs U where S u i , 0.2, 0.2 , 0.5, 0.5 : u i U and T
u , 0.3, 0.3 , 0.4, 0.4 i
: u i U . Consequently, we get that I Z JL ( S , T ) 0, IY (S , T ) 0
and I MT ( S , T ) 0, but S T . It implies that the measures (8)-(11) do not fulfill the axiom (D2) of Definition 3.5. Remark 3.2: Normally, consider
T u , m (u ), m (u ) m (u ) , n (u ), n (u ) n (u )
S ui , mS (ui ), mS (ui ) mS (ui ) , nS (ui ), nS (ui ) nS (ui ) : ui U , i
T
i
T
i
T
i
T
i
T
i
T
i
: ui U ,
Then I Z JL ( S , T ) 0, IY (S , T ) 0 and I MT ( S , T ) 0, but S T . Thus, the axiom (D2) of Definition 3.5 is violated for the divergence measures (8)-(11). 4. IVIF-Information Measures In this section, some existing information measures are discussed firstly. Next, new IVIFdivergence and IVIF-entropy measures are proposed, which are applied to the computation of the criterion weight in MCDM problems. 4.1. IVIF-Divergence Measure Due to the increase in complexity of the real-life problems, divergence measure for IVIFSs has attracted more attention of researchers in the decision-making field of study. However, existing information measures for IVIFS may be invalid, which will be explained later. To cope with such invalidity, a new divergence measure of order
for IVIFSs is proposed.
For S , T IVIFSs (U ), based on Ansari et al. (2018), a new IVIF-divergence measure is given by
p mS (ui ) 1 exp 1 mS (ui ) ln I S || T 1 1 2 ( ) ( ) m u m u 1 i S i T i
(13)
mS (ui ) nS (ui ) m (ui ) ln n u ( ) ln 1 2 mS (ui ) mT (ui ) S i 1 2 nS (ui ) nT (ui ) S
nS (ui ) 1 . nS (ui ) ln 1 2 nS (ui ) nT (ui )
Since 0 I S || T
1 1
1 41
1 . For simplicity, we normalize (13) and the cross-entropy
for IVIFSs is given by p mS (ui ) exp 1 ( ) ln I S || T m u S i 1 2 mS (ui ) mT (ui ) p 4 1 1 i 1 1
(14)
mS (ui ) nS (ui ) ( ) ln mS (ui ) ln n u 1 2 mS (ui ) mT (ui ) S i 1 2 nS (ui ) nT (ui ) nS (ui ) 1 . nS (ui ) ln 1 2 nS (ui ) nT (ui )
Now, the divergence measure for IVIFSs is given by
I S , T I S || T I T || S , p mS (ui ) 1 exp I S , T 1 m ( u ) ln S i 1 2 mS (ui ) mT (ui ) 2 p 4 1 1 i 1 mS (ui ) nS (ui ) ( ) ln n u mS (ui ) ln 1 2 mS (ui ) mT (ui ) S i 1 2 nS (ui ) nT (ui )
p nS (ui ) mT (ui ) exp 1 mT (ui ) ln nS (ui ) ln 1 2 nS (ui ) nT (ui ) 1 2 m ( u ) m ( u ) 1 i S i T i
(15)
mT (ui ) nT (ui ) m (ui ) ln u n ln ) ( 1 2 mS (ui ) mT (ui ) T i 1 2 nS (ui ) nT (ui ) T
nT (ui ) 2 . nT (ui ) ln 1 2 nS (ui ) nT (ui )
Theorem 4.1: Let S , T , W IVIFSs U and 1( 0). Then, I S , T fulfills the following postulates: (D1). I ( S , T ) I (T , S ); (D2). I ( S , T ) 0 iff S T ; (D3). I S , T I S c , T c and I S c , T I S , T c ; (D4). 0 I S , T 1; (D5). If S T W , then I S , T I S , W , I T , W I S , W . (D6). I ( S W , T W ) I ( S , T ), W IVIFSs U ; (D7). I ( S W , T W ) I ( S , T ), W IVIFSs U ; Proof: The proof is depicted in the Appendix. Remark 3.3: If IVIFSs transform to IFSs, then the divergence measure given by (13) will be converted to IF-divergence measure given by Ansari et al. (2018). The cross-entropy (14) is utilized to develop an entropy measure for IVIFSs. The above-mentioned examples 3.1 and 3.2 prove the advantage of the measure (15) over the others. The proposed measure (15) can be applied tothedecision-making process, edge detection, pattern recognition, etc. 4.2. IVIF-Entropy Measure Based on Vlachos and Sergiadis (2007), a relation between IVIF-divergence measure and IVIFentropy is developed. With the help of (15), we obtain
I S , S c 1
p mS (ui ) exp 1 m (u ) ln S i 1 2 mS (ui ) m c (ui ) p 4 1 1 i 1 S 1
mS (ui ) mS (ui ) ln 1 2 mS (ui ) mS c (ui )
nS (ui ) n (ui ) ln 1 2 nS (ui ) n c (ui ) S S
nS (ui ) nS (ui ) ln 1 2 nS (ui ) nS c (ui )
1 1.
It implies I S , S c 1
p mS (ui ) exp 1 m (u ) ln S i p 4 1 1 m u n u 1 2 ( ) ( ) i 1 S i S i 1
mS (ui ) nS (ui ) ( ) ln mS (ui ) ln n u 1 2 mS (ui ) nS (ui ) S i 1 2 nS (ui ) mS (ui ) nS (ui ) 1 u u nS (ui ) ln 1, ( ) ln 2 ( ) ln 2 4 1 S i S i 1 2 nS (ui ) mS (ui )
1
p 41
p mS (ui ) exp 1 m (u ) ln S i 1 m u n u 1 2 ( ) ( ) i 1 S i S i
mS (ui ) nS (ui ) ( ) ln mS (ui ) ln n u 1 2 mS (ui ) nS (ui ) S i 1 2 nS (ui ) mS (ui ) nS (ui ) 1 . nS (ui ) ln u u ( ) ln 2 ( ) ln 2 S i 1 2 nS (ui ) mS (ui ) S i
Thus, a relation between cross-entropy and entropy measures is obtained as follows:
H1 S 1 I S , S c , where
(16)
H1 S
1
p 41
p mS (ui ) exp 1 m (u ) ln S i 1 2 ( ) ( ) m u n u 1 1 i S i S i
(17)
mS (ui ) nS (ui ) m (ui ) ln n u ( ) ln 1 2 mS (ui ) nS (ui ) S i 1 2 nS (ui ) mS (ui ) S
nS (ui ) 1 . ( ) ln 2 ( ) ln 2 u u nS (ui ) ln S i 1 2 nS (ui ) mS (ui ) S i
Theorem 4.2: The measure H1 S , given by (17), is an IVIF-entropy. Proof: The proof is depicted in the Appendix. For each S IVIFS (U ), an IVIF-entropy is represented by H 2 S and given by
H 2 S
p
1 p
i 1
1 2 m in m S ( u i ), n S ( u i ) 1 m in m S ( u i ), n S ( u i ) S ( u i ) S ( u i )
2 m ax mS ( ui ), nS ( ui ) S i S i e S ( u i ) S ( u i ) 1 m ax m ( u ), n ( u )
12 max m S (u i ), n S (u i ) 2min mS ( ui ), nS ( ui ) S i S i S ( u i ) S ( u i ) max m S (u i ), n S (u i ) e . (u ) (u ) S i S i
1 min m ( u ), n ( u )
(18)
Theorem 4.3: The measure H 2 S , given by (18), is an IVIF-entropy. Proof: The proof is depicted in the Appendix. 4.3. Mathematical Comparison This section presents a comparison that illustrates the validity of proposed IVIF-entropy. Let S IVIFS (U ). For a real number p 0, De et al. (2000) introduced the set S p IVIFS (U ), which is given as follows:
(19)
p p p p S p ui , mR (ui ) , mR (ui ) , 1 1 nS (ui ) ,1 1 nS (ui ) : ui U .
Guo and Song (2014): H GS S
1 0.5 S ( ui ) S ( ui ) 1 p 1 1 m ( u ) n ( u ) m ( u ) n ( u ) , R i R i S i S i 2 p i 1 2
(20)
,
(21)
Wei and Zhang (2015):
H WZ
1 p mS ( ui ) nS ( ui ) mS ( ui ) nS ( ui ) S cos 2 2 S ( ui ) S ( ui ) p i 1
Meng and Chen (2015): 1 p min mS (ui ), nS (ui ) min mS (ui ), nS (ui ) , H MC S p i 1 max mS (ui ), nS (ui ) max mS (ui ), nS (ui )
(22)
Rani et al. (2018):
1 H R ( S ) p e 1 i 1 p
1 4
n
1 4
m
S
( u i ) m S ( u i ) 2
n S ( u i ) n S ( u i )
( u i ) n S ( u i ) 2 e m S (u i ) m S (u i ) S
1 4
m
S
( u i ) m S ( u i )
e
1 4
2
n S ( u i ) n S ( u i )
n
S
( u i ) n S ( u i )
2
m S ( u i ) m S ( u i )
(23)
1 .
Consider that S IVIFS U is defined by
S 6, 0.1, 0.2 , 0.6, 0.7 , 7, 0.2, 0.3 , 0.5, 0.6 , 8, 0.6, 0.7 , 0.1, 0.2 ,
(24)
9, 0.8, 0.9 , 0.0, 0.1 , 10, 1.0,1.0 , 0.0, 0.0. Taking into account the categorization of linguistic variables, De et al. (2000), Hung and Yang (2006), Xia and Xu (2012), and Rashid et al. (2018) considered S as “LARGE” on U . Using (19), we acquire
S 1 2 measured as “More or less LARGE.” S 2 measured as “Very LARGE.” S 3 measuredas “Quite very LARGE.” S 4 measured as “Very very LARGE.”
At this juncture, we calculate the entropy measures (17), (18), and (20)-(23), then the comparison outcomes are portrayed in Table 1. It is distinguished from the mathematical logical procedure that, IVIF-entropy measures have the following preference order:
H S1 2 H S H S 2 H S 3 H S 4 .
(25)
Table 1: Different IVIF-entropy values
S1 2
S
S2
S3
S4
H GS S
0.3551
0.2260
0.2042
0.1984
0.1864
HWZ S
0.5280
0.5460
0.4736
0.4206
0.3910
H MC S
0.3440
0.1950
0.1777
0.1983
0.1449
HR S
0.5321
0.5605
0.4718
0.4169
0.3870
H10.2 S
0.5102
0.4817
0.3562
0.3054
0.2675
H10.4 S
0.5396
0.5183
0.3908
0.3336
0.2949
H10.6 S
0.5913
0.5768
0.4479
0.3843
0.3444
H2 S
0.4755
0.4572
0.3976
0.3653
0.3344
In Table 1, the entropy measures except HWZ S , H MC S , and H RJ S satisfied the requirements mentioned in (25). The behaviors of IVIF-entropy measures H GS S , H1 S , and
H 2 S are thus reasonably good from the perspective of ordered linguistic variables. Remark 4.4: From the above-mentioned example, we can see that the entropy measures (17) and (18) successfully hold the order pattern given in (25), which shows their reliability. In this
paper, a correlative MCDM method is discussed with incomplete criteria weights information using proposed entropy measures. 5. Proposed IVIF-COPRAS Method for MCDM Here, the Shapley function-based COPRAS approach is developed to tackle the correlative MCDM problems under the IVIF doctrine. 5.1. Proposed IVIF-COPRAS Method with Shapley Function The Complex Proportional Assessment (COPRAS) approach pioneered by Zavadskas et al (1994) is extremely useful for solving MCDM problems. The key outcomes of the COPRAS are as follow (Mishra et al., 2018; Zheng et al., 2018): first, it can simultaneously take the ratios to the optimal and the worst solutions. Second, outcomes can be achieved in a very less time as compared to the different existing MCDM methods. Finally, it is computationally straightforward and easily comprehensible. Based on the benefits, the COPRAS approach has been widely implemented in various decision-making enviroments. In the conventional COPRAS approcah, the assessment of alternatives over the criteria is distinguished as crisp numbers. Nonetheless, in diverse circumstances, crispness is not enough to handle real-life MCDM problems. As IVIFSs have been widely implemented to solve the uncertain MCDM problems, the conventional COPRAS method is generalized. The procedural steps of IVIF-COPRAS approach are depicted as follows (graphically drawn in Fig. 1): Step 1: Create an IVIF-decision matrix.
For an MCDM problem, generate a set of alternatives B B1 , B2 ,..., B p and classify the set of
criteria G G1 , G2 ,..., Gq . The decision expert presents his/her assessment values i j of the alternatives B i i 1(1) p with respect to criteria G j j 1(1) q in the form of linguistic variables (LVs). Next, we transform LVs into IVIFNs, which constructs an IVIF-decision matrix M ij
pq
, where ij ij , ij , ij , ij , for experts.
Step 2: Assess the DEs weights
Let k 1 , 2 ,..., be the weight values of DEs. These DEs weights are considered as T
linguistic values and expressed in IVIFNs. Let ek mk , mk , nk , nk be a rating of the k th DE in terms of IVIFNs. According to Boran et al. (2009), the DEs’ weight is evaluated by mk mk m k mk k k mk nk mk nk , k 1, 2,..., . k m m mk k k mk k k m n k 1 k k mk nk
Here, k 0 and
l
(26)
k 1.
k 1
Step 3: Find the aggregated IVIF (A-IVIF) decision matrix
Let M ij k
k
be the decision matrix of k th DE, where ij mijk , mijk , nijk , nijk ,
k 1, 2,..., is IVIFNs. Consequently, to combine all the individual matrices and generate a
combined decision matrix, an A-IVIF decision matrix is constructed. To do this, let zij p q where zij aij , bij , cij , dij , i 11 p, j 11 q be the A-IVIF decision matrix, where k 1 k z ij k and l
l l l l k zij 1 1 mijk ,1 1 mijk k , nijk k , nijk k . k 1 k 1 k 1 k 1
(27)
Step 4: Evaluate the criteria weights. To distinguish the importance of the criteria, we have to compute the criterion weight. For correlative characteristics among the criteria, the weight vector is computed as follows: Step 4.1: Compute the entropy value of alternative Bi associated with criteria G j . A linear programming model (LP-model) for desirable fuzzy measure associated with criteria
G is developed when the information associated with criteria weights is completely unknown:
q
(28)
p
min H zij j , G j 1 i 1
0, G 1, s. t. L M L, M G , L M ,
where j , G is the Shapley value of criteria G j j 1 1 q and H zij is an entropy value for the IVIFN zij . Whereas, if the information associated with the criteria weights is partially known, then the LPmodel is assembled as follows: q
(29)
p
min H z ij j , G j 1 i 1
0, G 1, s. t . G j Y j , j 1(1) q , L M , L , M G , L M ,
where H zij is the entropy for the IVIFN zij and Y j y j , y j is its range. Utilizing (7), estimate the Shapley values of each criterion. Therefore, the Shapley values are employed as the criteria weights for the correlative MCDM problem. Step 5: Determine the normalized aggregated IVIF decision matrix. Normalize
the
A-IVIF
decision
matrix
zij
mn
into
N ij
p q
,
where
ij ij , ij , ij , ij is IVIFN, using the formula
ij
ij
aij
12
2 2 p aij bij i 1
cij
2 2 p cij dij i 1
, ij
bij
12
2 2 p aij bij i 1
,
dij , , ij 12 1 2 i 11 p, j 11 q. 2 2 p cij dij i 1
(30)
(31)
Figure 1: Framework of the IVIF-COPRAS method for MCDM
Step 6: Sum the criteria values for benefit-type and cost-type. In the proposed approach, each option is illustrated with its additions of maximizing criterion,
i , which is assigned to benefit type and minimizing criterion, i , which is assigned to cost type.
To evaluate the value of i and i , we implement the following procedure: s
i j , G ij , i 11 p,
(For benefit-type)
(32)
j 1 q
, G
i
j s 1
j
ij
, i 11 p.
(For cost-type)
(33)
In formulas (32) and (33), s describes the maximizing (benefit-type) criteria, q shows the entire criteria, and j , G is importance degree of the jth criteria. Step 7: Evaluate the relative degree of the alternative. The relative degree or weight i of option is evaluated by
i * i
(34)
p
min i * i *
i
i 1
p
* i
min * i
, i 11 p
i
* i
i 1
where * i and * i represent the score values of i and i , respectively. Next, we can also inscribe (34) as follows: p
i * i
i
i 1
p
i *
(35)
*
i 1
1
, i 11 p.
* i
Step 8: Obtain the preference order or degree. The preference degree of alternatives is evaluated according to the relative degree or weight of the alternative. The maximum relative weight determines the higher preference order (rank) of the alternative, which is the desirable one. Thus, B* max i , i 11 p. i
(36)
Step 9: Compute the utility degree. The utility degree i is computed by comparing the evaluated alternative with the optimal one. The range of the i lies from 0% to 100% representing the worst and the optimal ones, respectively. Therefore, the utility degree i of each alternative is calculated by
i
i 100%, i 11 p, max
(37)
where i and max are the relative degrees of the alternative. The proposed MCDM method allows us to evaluate the straight and comparative dependence of the relative degree and utility degree of alternatives associated with the criteria set.
5.2. Evaluation of HWR Facility Selection Problem using the IVIF-COPRAS method This section discusses an MCDM problem of selecting the HWR facility for the purpose of verifying the efficiency of the proposed IVIF-COPRAS approach. Here is presented the results obtained from a pilot study carried out for the Indian Health Service (IHS) in the Albuquerque region. It was aimed to assessthehealth and safety status in the case of chosen HWR facilities through the framework introduced in the present study. IHS formally provides American Indians and Alaska Natives with federal health services. Suchestablishment of health services for the members of tribes, which are federally-recognized,is actually originated from the relationships between thesetribes and the federal government. The Albuquerque area is broken out by administrativeregions of service units. The formal responsibility of IHS is improvingthe American Indians and Alaska Natives physically, mentally, socially, and spiritually to utmost levels. It attempts to provide healthy communities and healthcare systems of high-quality throughrobust partnerships and culturally-responsive practices. IHS is mainly aimed at ensuring that all-inclusive, culturally-proper personal, and public health services are completely available for American Indians and Alaska Natives; promoting a high quality of lifeby innovating the Indian health system into an optimally performing organization; and also strengtheningthemanagement and operations of IHS. Five alternative facilities within the Albuquerque area were chosen to take part in the present research through the IHS officials; they are Jicarilla Service Unit ( B1 ), Ute Mountain Ute Service Unit ( B2 ), Zuni Comprehensive Health Center ( B3 ), New Sunrise Regional Treatment Center ( B4 ), and Taos-Picuris Service Unit ( B5 ). One of the seasoned inspectors was chosen by the IHS officials to lead this effort. The lead inspector E1 invited a colleague from the Division of Facilities Planning Construction (DFPC) E2 and another colleague from the Division of Environmental Health Services (DEHS) E3 to join him in this collaborative initiative. The three
experts E1 , E2 , E3 of the evaluation team (i.e., the IHS inspector, the DFPC inspector, and the DEHS inspector)were chosen to make an evaluation on the HWR Facility option. In addition, the assessment team agreed to employ the linguistic variables listed in Tables 2 and 3 in the description of the important weights and the performance ratings of the criteria, respectively. The team took into account the following four criteria (which were established by IHS) for the purpose of assessing in asystematic way the potential safety and health hazards in waste recycling facilities all through the Albuquerque area: severity of occurrence ( G1 ), time of exposure ( G2 ), failure to detect the risk ( G3 ), and protective & preventive measures ( G4 ). Here, the criteria G1 , G2 , G3 are the benefit criteria and G4 is the cost criterion, and the scale of the criteria is specified by
0.4, 0.6 , 0.1, 0.3 , 0.15, 0.3 and 0.15, 0.45. Now,
the procedure for execution of IVIF-
COPRAS method is given by Step 1: Constructionof an IVIF-decision matrix. Each decision expert executes his/her opinions regarding the rating of HWR selection over the preferred criteria. Here, Table 2 represents the linguistic variables in the form of IVIFNs to estimate the relative importance of preferred assessment criteria. Table 3 expresses the preference of decision experts
E1 , E2 , E3
for the criterion weight. Based on the IVIFN scale, Table 4
presents the linguistic values for the performance of the HWR facility evaluated by decisionmakers. Table 5 expresses the individual decision opinions of each Ei concerning the performance of HWR selection. Table 2: Importance weights as linguistic terms Linguistic variables
IVIFNs
Very important (VS)
0.80, 0.95 , 0.0, 0.05
Important (S)
0.65, 0.75 , 0.15, 0.20
Modest (M)
0.45, 0.55 , 0.30, 0.45
Unimportant (IS)
0.20, 0.30 , 0.55, 0.70
Very unimportant (VIS)
0.00, 0.10 , 0.80, 0.90
Table 3: Importance weights of
Ei i 1, 2,3 as linguistic terms
E1
E2
E3
Very important
Modest
Important
IVIFNs
0.80,0.95 ,0.0,0.05
0.45,0.55 , 0.35,0.45
0.65,0.75 ,0.15,0.20
Weights
0.394
0.274
0.332
Linguistic values
Table 4: Significance rating of HWR facility option as linguistic values Linguistic terms
IVIFNs
Extremely good (XG)/ extremely high (XH)
0.90, 0.95 , 0.00, 0.05
Very very good (EEG)/ very very high (EEH)
0.80, 0.90 , 0.05, 0.10
Very good (EG)/ very high (EH)
0.70, 0.80 , 0.15, 0.20
Good (G)/ high (H)
0.60, 0.70 , 0.20, 0.25
Fairly good (FG)/ moderate high (OH)
0.50, 0.60 , 0.25, 0.30
Fair (F)/ moderate (O)
0.40, 0.50 , 0.35, 0.40
Fairly bad (FB)/ moderate low (OL)
0.30, 0.40 , 0.45, 0.50
Bad (B)/ low (L)
0.15, 0.25 , 0.55, 0.60
Very bad (EB)/ very low (EL)
0.10, 0.15 , 0.70, 0.75
Very very bad (EEB)/ very very low (EEL)
0.00, 0.10 , 0.85, 0.90
Table 5: Performance rating of HWR options from DEs as linguistic terms HWR alternative
B1
B2
B3
DEs
G1
G2
G3
G4
E1
OH
EEG
G
O
E2
H
FG
FB
OH
E3
OL
G
B
H
E1
EH
FB
FB
EEL
E2
OH
EEG
G
L
E3
O
EB
EB
EL
E1
O
G
EEG
EH
E2
OL
F
XG
O
E3
L
FB
EG
H
B4
B5
E1
H
FG
F
OL
E2
O
B
EG
OL
E3
OH
F
G
EL
E1
OH
XG
EG
EH
E2
H
EG
G
OH
E3
OL
EEG
EEG
H
Based on Tables 3-5 and Eq. (27), the A-IVIF-decision matrix is constructed and given in Table 6. Table 6: A-IVIF-decision matrix for HWR facility alternative
G1
G2
G3
G4
B1
0.406, 0.521 , 0.232, 0.330
0.614, 0.776 , 0.134, 0.204
0.398, 0.522 , 0.195, 0.418
0.517, 0.699 , 0.124, 0.208
B2
0.490, 0.732 , 0.098, 0.212
0.332, 0.778 , 0.122, 0.200
0.287, 0.403 , 0.375, 0.582
0.138, 0.217 , 0.440, 0.520
B3
0.267, 0.324 , 0.583, 0.664
0.4, 0.5 , 0.333, 0.420
0.707, 0.794 , 0.121, 0.201
0.552, 0.618 , 0.237, 0.359
B4
0.505, 0.614 , 0.131, 0.235
0.322, 0.435 , 0.226, 0.313
0.544, 0.758 , 0.128, 0.208
0.151, 0.201 , 0.677, 0.781
B5
0.481, 0.542 , 0.354, 0.427
0.830, 0.919 , 0.022, 0.072
0.621, 0.781 , 0.113, 0.197
0.561, 0.631 , 0.131, 0.261
Step 2: Computation of the criterion weight. To evaluate the criterion weight, (i)
The entropy measure (18) of each HWR facility Bi i 11 5 is computed associated with
criteria G j i 11 4 in Table 7. Table 7: Entropy values of A-IVIF-decision matrix for HWR facility
G1
G2
G3
G4
B1
0.8716
0.5728
0.8903
0.6683
B2
0.6588
0.7205
0.9006
0.8001
(ii)
B3
0.7350
0.9453
0.5006
0.7743
B4
0.7280
0.9303
0.6225
0.5373
B5
0.9052
0.2451
0.5585
0.6978
The LP- model (28) for the optimal fuzzy measure is constructed as follows:
min [0.1110{ (G1 ) (G 2 , G3 , G 4 )} 0.0506{ (G 2 ) (G1 , G3 , G 4 )} 0.0311{ (G3 ) (G1 , G 2 , G 4 )} 0.0293{ (G 4 ) (G1 , G 2 , G3 )} 0.0302{ (G1 , G 2 ) (G3 , G 4 )} 0.0399{ (G1 , G3 ) (G 2 , G 4 )} 0.0408{ (G1 , G 4 ) (G 2 , G3 )} 3.5657] (G1 ) (G1 , G2 ), (G1 ) (G1 , G3 ), (G1 ) (G1 , G4 ), (G2 ) (G1 , G2 ), (G2 ) (G2 , G3 ), (G2 ) (G2 , G4 ), (G3 ) (G1 , G3 ), (G3 ) (G2 , G3 ), (G ) (G , G ), (G ) (G , G ), (G ) (G , G ), (G ) (G , G ), 3 3 4 4 1 4 4 2 4 4 3 4 (G1 , G2 ) (G1 , G2 , G3 ), (G1 , G3 ) (G1 , G2 , G3 ), (G2 , G3 ) (G1 , G2 , G3 ), s. t. (G1 , G2 ) (G1 , G2 , G4 ), (G1 , G4 ) (G1 , G2 , G4 ), (G2 , G4 ) (G1 , G2 , G4 ), . (G , G ) (G , G , G ), (G , G ) (G , G , G ), (G , G ) (G , G , G ), 1 3 1 3 4 1 4 1 3 4 3 4 1 3 4 (G2 , G3 ) (G2 , G3 , G4 ), (G3 , G4 ) (G2 , G3 , G4 ), (G2 , G4 ) (G2 , G3 , G4 ), (G1 , G2 , G3 ) 1, (G1 , G3 , G4 ) 1, (G1 , G2 , G4 ) 1, (G2 , G3 , G4 ) 1, (G ) 0.4, 0.6 , (G ) 0.1, 0.3 , (G ) 0.15, 0.3 , (G ) 0.15, 0.45 . 1 2 3 4
(38)
Using MATHEMATICA, solve (38) to compute the fuzzy measure associated with criteria G thatis given by
(G1 ) 0.4, (G2 ) 0.3, (G3 ) 0.3, (G4 ) 0.4 (G1 , G2 ) (G1 , G3 ) (G1 , G4 ), (G2 , G3 ) 1 (G2 , G4 ), (G3 , G4 ) 0.4, (G1 , G3 , G4 ), (G1 , G2 , G3 ) 1 (G1 , G2 , G4 ) (G2 , G3 , G4 ) (G1 , G2 , G3 , G4 ). (iii) The
calculated
Shapley
values
are
GH , G 0.1167, GH , G 0.4834, 2
1
2
2
GH , G 0.1833 and GH , G 0.2166. 2
3
2
4
Step 3: Creation ofa Normalized A-IVIF-decision matrix. Using formulas (30) and (31), the normalized A-IVIF-decision matrix is computed and given in Table 8. Table 8: Normalized A-IVIF-decision matrix
G1
G2
G3
G4
B1
0.1166, 0.1497 , 0.0990, 0.1408
0.1443, 0.1824 , 0.0908, 0.1382
0.0960, 0.1259 , 0.1049, 0.2248
0.1695, 0.2291 , 0.0463, 0.0776
B2
0.1408, 0.2103 , 0.0418, 0.0904
0.0780, 0.1828 , 0.0867, 0.1355
0.0692, 0.0972 , 0.2017, 0.3130
0.0452, 0.0711 , 0.1642, 0.1941
B3
0.0767, 0.0931 , 0.2487, 0.2832
0.0940, 0.1175 , 0.2257, 0.2846
0.1706, 0.1916 , 0.0651, 0.1081
0.1809, 0.2026 , 0.0885, 0.1340
B4
0.1451, 0.1764 , 0.0559, 0.1002
0.0757, 0.1022 , 0.1531, 0.2121
0.1312, 0.1829 , 0.0688, 0.1119
0.0495, 0.0659 , 0.2527, 0.2915
B5
0.1382, 0.1557 , 0.1510, 0.1821
0.1951, 0.2160 , 0.0149, 0.0488
0.1498, 0.1884 , 0.0608, 0.1059
0.1839, 0.2068 , 0.0489, 0.0974
Here, the severity of occurrence ( G1 ), time of exposure ( G2 ), failure to detect the risk ( G3 ), and protective & preventive measures ( G4 ) are given criteria to evaluate HWR facility alternative selection problem. As mentioned earlier, the criteria G1 , G2 and G3 are the benefit criteria and G4 is the cost criterion. Using (32)-(37), the values of i , * i , i , * i , i and i of
Bi i 11 5 are calculated related to criteria G j j 1 1 4 , which are given inTable 8. Table 9: The computational outcome of IVIF-COPRAS approach for HWR facility selection Alternatives
i
* i
i
* i
i
i
B1
0.1026,0.1315 , 0.1584,0.2325
0.4608
0.0394, 0.0548 , 0.5140, 0.5748
0.2514
0.6094
83.15%
B2
0.0677,0.1340 , 0.1579,0.2323
0.4529
0.0100,0.0158 , 0.6762,0.7011
0.1621
0.6834
93.25%
B3
0.0873,0.1049 , 0.2509,0.3127
0.4071
0.0423,0.0479 , 0.5914,0.6470
0.2129
0.5826
79.49%
B4
0.0788,0.1058 , 0.1765,0.2418
0.4416
0.0109,0.0147 , 0.7423,0.7657
0.1294
0.7329
100%
B5
0.1410,0.1611 , 0.0628,0.1262
0.5283
0.0431,0.0489 , 0.5201,0.6038
0.2420
0.6827
93.15%
From Table 9, the preference order of HWR facility selection is B4 B2 B5 B1 B3 ,thus, B4 is the best HWR facility. 5.3. Sensitivity Analysis In this paper, a sensitivity analysis was done to investigate the proposed approach behavior. Four different criteria weight sets are taken and depicted in Table 10. In this table, for every set, one of the criterion has the maximum weight, while the other ones have lower weights. By applying this process, an ample scope of criteria weights was created to investigate the sensitivity ofthe developed approach to variation of the criteria weights.
Figure 2: Results of
i for each alternative with different values of weight sets
Table 10: Different criteria weight sets for HWR facility alternative Weight sets
G1
Set- I Set – II
G2
G3
G4
0.1167
0.4834
0.1833
0.2166
0.2166
0.1167
0.4834
0.1833
Set – III
0.1833
0.2166
0.1167
0.4834
Set – IV
0.4834
0.1833
0.2166
0.1167
The sensitivity analysis results described in Fig. 2 showed that the relative degree i couldchange over different criteria weight sets and the rank of HWR facility alternative. For example, when DEs give the weight set-I, III, and IV, the ranking of HWR facility ranking is
B4 B2 B5 B1 B3 , whereas when the evaluation weight set-II is taken, then HWR facility ranking becomes B4 B2 B5 B3 B1. From the above discussion, it is concluded that the HWR facility selection is dependent on and sensitive to these criteria weight sets. Therefore, the proposed approach has acceptable stability with different weight sets. 5.4. Comparison and Discussion In this section, we first compare the proposed IVIF-COPRAS approach with the IVIF-TODIM adopted from Mishra and Rani (2018) method in the same environment (an example is presented in Section 5.1). Furthermore, a comparison is made between the proposed and IVIF-TODIM approaches. In the same environment, the classical TODIM method is implemented, in which the evaluation values are given in the form of IVIFNs. In the following, the computation steps for the TODIM method are discussed. Step 1: Determining the relative weight vector. The relative weight vector jr ( , G ) of G j associated withcriterion Gr is calculated using the formula jr , G
j , G , where r , G max j , G and j , G is the criteria r , G
weight. Step 2: Calculating the dominance matrix. The degree of dominance of alternative Bi associated withan alternative Bt is computed using the expression
jr , G I ( zij , ztj ) if zij ztj , q jr , G j 1 j Bi , Bt if zij ztj , 0, q 1 jr , G I ( ztj , zij ) j 1 , if zij ztj jr , G
(39)
where the term represents the reduction factor that controls the impact of loss, I ( zij , ztj ) describes the degree of dissimilarity between IVIFNs zij and ztj , and jr , G shows the criterion Shapley value. Step 3: Evaluating the overall dominance degree. The overall dominance degree j Bi , Bt of alternative Bi over Bt is evaluated using the expression (40)
q
Bi , Bt j Bi , Bt , i, t 1(1) p. j 1
Step 4: Determining the overall prospect or global value. Based on the conventional TODIM method, we determine the overall dominance degree of alternative Bi associated with alternative Bt as follows:
B , B min B , B , i 1(1) p. B max B , B min B , B p
p
i
k 1
i
t
i
p
i
k 1
i
k 1
t
(41)
p
i
t
i
k 1
i
t
Step 5: Considering the preference degree of the options and choosing the optimal one. Preference order of the alternatives based on the increasing order of global values and choose the optimal one. The higher overall prospect value determines the optimal alternative. The results obtained by applying the IVIF-TODIM method to the HWR facility selection problem mentioned in Section 5.1 are given as below: Step 1: The relative weight vector is evaluated as 1r ( , G ) 0.2414, 2 r ( , G ) 1.0000,
3r ( , G ) 0.3792 and 4r (, G) 0.4481.
Step 2: Through the use of Eq. (39), the dominance degree of each HWR alternative
Bi i 11 5 is calculated under the criteria G j j 11 4 at 2.5; thus, the dominance matrix is expressed as follows: 0 0.2014 1 0.1953 0.1433 0.0809
0.1232 0 0.3125 0.0442 0.2818
0.3195 0.0876 0.0809 0.5110 0.0723 0.4610 0.2792 0.1649 , 0 0.4567 0 0.2164 0.2697 0.1323 0
0.0705 0.1426 0.1209 0 0.1828 0.1344 0.1073 0 2 0.3698 0.3487 0.0956 0 0 0.3136 0.2782 0.0368 0.1220 0.1578 0.2509 0.2812 0 0.2965 3 0.1059 0.0878 0.0583
0.0772 0 0.1782 0.1592 0.2055
0.3166 0.4092 0.9266 , 0.7294 0
0.4071 0.3373 0.2239 0.6850 0.6117 0.7896 0.2194 0.0125 , 0 0.8433 0.0632 0 0.0480 0.0165 0
0.1841 0.0102 0 0.8415 0.6760 0 4 0.0468 0.1479 0 0.2263 0.2014 0.8564 0.1012 0.1753 0.0410
0.2283 0.0221 0.0441 0.8010 0.1874 0.1873 . 0.9895 0 0 0.2165
Step 3: From Eq. (40), the overall dominance degree of each HWR alternative Bi i 11 5 is determined under the criteria G j j 11 4 , therefore, the overall dominance matrix is articulated as below: 0.2086 0.0652 0.0777 0.4375 0 1.1194 0 0.7156 0.3880 1.5388 0.5060 0.3351 0 0.0320 1.2663 . 0 0.7635 0.1438 0.3646 1.2062 0.1600 0.2568 0.5136 0.3819 0
Step 4: Through the use of Eq. (41), the overall prospect value Bi of each HWR facility option Bi is determined as follows:
B1 0.6938, B2 0.0000, B3 0.3097, B4 1.0000 and B5 0.3324. Step 5: Corresponding to Step 4, the ranking of the HWR is B4 B1 B5 B3 B2 ; thus, B4 is the best HWR alternative. According to the prospect theory, Kahneman and Tversky (2013) suggested that the factor of attenuation of losses is around 2.25, and [1, 2.5]. To observe the impact of the factor in the problem;various factor values were implemented from 1 to 2.5; then, the same ranking outcomes were achieved. Consequently, the HWR option B4 is the most desirable one for any value of . The outcome of IVIF-TODIM method not only determines the most desirable HWR alternative but also showsthe psychosomatic behaviors of the DEs. In this section, to illustrate the superiority of the IVIF-COPRAS approach, a comparison is also made between its results and those of the existing ones.In recent years, various decision-making approaches with different characteristics have been introduced for the safety and health evaluation in hazardous waste recycling context. Here, for the comparative study, we preferred the approaches that could be efficiently applied to the considered MCDM problem. Then, the approaches proposed by Hashemi et al. (2015), Kuo et al. (2015), and Tang (2017) were extracted from literature for the analysis purposes. To have an effective comparison, the Spearman's rank correlation rB between the developed approach and existing ones were calculated (see Table 12). As can be observed in Table 12, the correlation coefficients are greater than 0.6 (there is a high degree of dependency between ranking outcomes). As a result, the relationships between preference order outcomes are strong or very strong. Based on the research findings, the outcome of the developed approach was found consistent with the existing approaches, except the method of Kuo et al. (2015) and the IVIF-TODIM (Mishra and Rani (2018)). According to Tables 11 and 12, the methods developed by Hashemi et al. (2015), Kuo et al. (2015), and Tang (2017), as well as the IVIF-TODIM (Mishra and Rani (2018)) have rankings different from that of our proposed method (Except for best HWR facility, B4 ). The approach developed by Hashemi et al. (2015), which is based on ANP-Grey relational analysis (GRA), is easy to implement; however, it doesnot consider the simplification in the weight assessment
process, lacks a basis to ensure for feasible inconsistencies, and ignores the inherent interdependencies with the criteria. Kuo et al. (2015) developed a hybrid model integrating DANP and VIKOR methods, which provided a compromise solution, but it was unable to consider the bounded rationality of decision-makers. The method presented by Tang (2017) is based on the aggregation operator. Its drawback is that different aggregation operators typically provide different rankings. However, the proposed IVIF-COPRAS method is developed to avoid the shortcomings of existing methods and provide more reasonable results. Table 11: Comparative outcomes of ranking order with various approaches Approaches
Benchmark
Environment
Rank order
Optimal HWR Facility
Kuo et al. (2015)
DANP based VIKOR
Crisp Sets
B4 B5 B1 B3 B2
B4
FSs
B4 B1 B5 B2 B3
B4
methods Hashemi et al.
ANP and Improved GRA
(2015)
methods
Keshavarz
Extended WASPAS method
IT2FSs
B4 B5 B2 B1 B3
B4
Hesitant fuzzy Hamacher
HFSs
B4 B5 B1 B2 B3
B4
IVIFSs
B4 B1 B5 B3 B2
B4
IVIFSs
B4 B2 B5 B1 B3
B4
Ghorabaee et al. (2014) Tang (2017)
power weighted average (HFHPWA) Operator method Mishra and Rani
Divergence measure based
(2018)
TODIM method
Proposed IVIF-
Divergence measure based
COPRAS method
improved COPRAS method
Table 12: Ranking of the HWR facility with existing approaches and coefficient of correlation rB HWR facility
Kuo et al. (2015)
B1
Hashemi et al. (2015) 2
Tang (2017)
Mishra and Rani (2018)
Proposed IVIFCOPRAS
3
Keshavarz Ghorabaee et al. (2014) 4
3
2
4
B2
4
5
3
4
5
2
B3
5
4
5
5
4
5
B4
1
1
1
1
1
1
B5
3
2
2
2
3
3
Spearman’s Correlation rB
0.60
0.40
0.90
0.70
0.30
-
On the basis of comparison with the IVIF-TODIM and other existing approaches, the key outcomes of the developed IVIF-COPRAS approach are as follow: (i)
The IVIF-COPRAS method proposed in this studycan easily handle various practical problems such as plant location problem, composite optimization, AHP-based decision making, radio frequency identification (RFID) network planning, and MAGDM. Whereas the existing methods and IVIF-TODIM cannot effectively perform dissimilarity assessment when the preference points are too large. Therefore, these methods are unable to handle large MCDM problems directly.
(ii)
In the IVIF-COPRAS method, the improved accuracy and score functions are used to tackle the preference attitude of the decision-maker and to make the ranking results more flexible than the other methods.
(iii) In the DANP-based VIKOR method introduced by Kuo et al. (2015), crisp values are used to estimate the HWR facility option. Indeed, the crisp numbers cannot properly model the uncertainty, ambiguity, and incompleteness in uncertain safety and health conditions in the HWR process. In contrast, the proposed IVIF-COPRAS approach is used to rank the HWR option completely inthe IVIF environment, which makes the safety and health assessment in the HWR process more practical and accurate. (iv) The approaches proposed by Hashemi et al. (2015), Kuo et al. (2015), Tang (2017), and the IVIF-TODIM approach have employed only maximizing criteria,and minimizing criteria must be converted into the maximizing criteria before implementation, i.e., the cost-type criteria must be transformed into benefit-type criteria before normalization. However, this procedure is not a minor task. For instance, the transformation in AHP may be the reason for conflicting outcomes, while in the proposed IVIF-COPRAS approach, this deficiency is removed. This is the superiority of IVIF-COPRAS over the others.
Figure 3: Comparison of preference order of HWR facility derived by the proposed method and the existing methods
A comparison between the preference order outcomes derived by the IVIF-COPRAS and the other methods shows a high correspondence isillustrated in Table 12 and Fig. 3. This implies that the ranking deduced by the IVIF-COPRAS should be considered as correct.Thus, the proposed IVIF-COPRAS approach can perform more reliably in handling the correlative MCDM problems. It can also be applied to the MCDM problems in which criteria are independent and have equal importance. 6. Conclusions IVIFSs are appropriate tools to articulate and handle the uncertain and imprecise information in many MCDM problems. The IVIF-information measures have been used in numerous fields such as decision-making, image processing, and medical diagnosis. This paper presented thenovel IVIFdivergence and IVIF-entropy measures based on the exponential function. Comparative examples were presented to illustrate the usefulness of the developed IVIF-information measures over the existing ones. Next, the conventional COPRAS approach was extended to calculate the correlative MCDM problems with IVIFSs. In the present method, the proposed IVIF-information measures are used to compute the criteria weights in the form of Shapley values. Furthermore, some adjustments were
done in the normalization procedure of the conventional COPRAS method. Then, an application related tosafety and health assessment in HWR facility selection wasutilized to illustrate the validity and usefulness of the proposed approach with IVIFSs. To validate the outcomes of the proposed method, a comparison was made between the developed IVIF-COPRAS and different existing approaches. The results confirmed that the developed approach was well-organized, easily applicable, andefficient. The advantages of the IVIF-COPRAS approach are the computation simplicity in IVIFSs, implementation of a process for calculating more realistic criteria weights, and increasing the stability of the approach. In the future, the developed MCDM model can be further extended to PFSs, interval-valued Pythagorean fuzzy sets, HFSs and interval-valued hesitant fuzzy sets. Moreover, the authors can expand our research using different MCDM platforms (for instance, AHP, Combined Compromise Solution (CoCoSo), ELECTRE, and MABAC) in order to select the most appropriate HWR facility selection. Appendix A A.1. Proof of Theorem 4.1 (D1). It is evident from Definition 3.5 that I S , T I T , S . (D2). It can easily be verified that I S, T 0 iff S T . (D3). From (13), it is also evident that I S , T I S c , T c and I S c , T I S , T c . (D4). As 0 mS (ui ), nS (ui ) 1, 0 ( 1), then, we have I S , T 0 , and I S , T 0 iff S T . The
I S , T attains
measure
the
maximum
value
when
m (u ), m (u ) , n (u ), n (u ) 1,1 , 0, 0 and m (u ), m (u ) , n (u ), n (u ) 0,0 ,1,1 , S
i
S
i
S
i
S
i
T
i
T
i
T
i
T
i
which follows that I S , T p ln 4. When T S c , then I S , T attains its maximum value and when S T , then I S , T has minimum value, therefore, the value of proposed divergence measure lies between p ln 4. For the process of normalization, we divided the I S , T by p ln 4 , we obtain 0 I S, T 1.
(D5). Consider S T W , then m S (u i ) mT (u i ) mW (u i ), m S (u i ) mT (u i ) mW (u i ),
nS (ui ) nT (ui ) nW (ui ) and nS (u i ) nT (u i ) nW (u i ). Now,
m (u ) 1 nS (ui ) mT (ui ) 1 nT (ui ) 0 S i 2 2 1 i
m (u ) 1 nS (ui ) mW (ui ) 1 nW (ui ) S i 1, 1( 0). (A1) 2 2 2 i
Let i exp i , then 1i 2i .
(A2)
This implies I S , T 1i 2i I S , W . i
i
Similarly, I T , W I S , W . (D6). To prove (D6), we partition U into eight subsets as follows: U ui U | S ui T ui W ui ui U | S ui W ui T ui ui U | S ui T ui W ui ui U | S ui W ui T ui ui U | T ui R ui W ui ui U | T ui W ui S ui ui U |W ui S ui T ui ui U |W ui T ui S ui , (A3)
which are denoted by 1 , 2 ,..., 8 . From Montes et al. (2002), for each j ; j 11 8,
and
S W ui T W ui S ui T ui
(A4)
S W ui T W ui S ui T ui .
(A5)
Therefore, from (D5), we obtain
I ( S W , T W ) I ( S , T ) for every W IVIFSs (U ). (D7). The proof is similar to (D6). A.2. Proof of Theorem 4.2
To show (24) a valid measure of IVIF entropy, it must hold the postulates (EN1)-(EN4) of Definition 3.4. The measure (18) can be written as
H1 S H S , (A6) where
H S
p 1 m ( u ) ln i S p ln 4 i 1
nS (ui )ln
nS ( ui ) mS ( ui ) nS ( ui )
m S ( u i ) m S
( ui )
n S
( ui )
n (u )ln S
i
m
S
( u i ) ln
nS ( ui ) mS ( ui ) nS ( ui )
m S ( u i ) m S
( u i ) n S ( u i )
(u )ln 2 (u )ln 2 S
i
S
i
(A7)
is an entropy measure introduced by Chen et al. (2010) and function ui : 0,1 [0,1] is defined as
ui p 41 1 exp 1 ui 1 .
(A8)
Here, function ui is increasing at ui 0. Therefore, H1 S is an increase of H S and H S is the IVIF-entropy. Hence, it is lucid that H1 S is the valid parametricIVIF-entropy. A.3. Proof of Theorem 4.3 To prove (18) asa valid measure of IVIF-entropy, it must hold the postulates (EN1)-(EN4) given in Definition 3.4. (EN1). Suppose that S be a crisp set, i. e., mS (ui ), mS (ui ) 1,1 , nS (ui ), nS (ui ) 0, 0 or
mS (ui ), mS (ui ) 0, 0 , nS (ui ), nS (ui ) 1,1 , for each ui U . Therefore, H 2 S 0. If H 2 S 0, then each term of (18) is non-negative. It implies that it should be equal to zero, i.
e., min mS (ui ), nS (ui ) 0, min mS (ui ), nS (ui ) 0, S (ui ) 0 and S (ui ) 0, for each ui U . Thus, S is a crisp set. (EN2). Let mS (ui ), mS (ui ) nS (ui ), nS (ui ) , i. e., mS (ui ) nS (ui ), mS (ui ) nS (ui ), for all ui U . From (18), we get H2 (S ) 1. Now, assume that H 2 ( S ) 1, then for all ui U , we have
1 max m ( u ), n ( u ) 1 S i S i 2 min ( ), ( ) m u n u max m ( u ), n ( u ) S i S i 2 S i S i p S ( ui ) S ( ui ) 1 min mS (ui ), nS (ui ) e p 1 i 1 S (ui ) S (ui )
1 min m ( u ), n ( u ) 12 max mS (ui ), nS (ui ) 2min mS (ui ), nS (ui ) S i S i S ( ui ) S ( ui ) max mS (ui ), nS (ui ) e 1. (u ) (u ) S i S i
1 12 min mS (ui ), nS (ui ) 2 max mS(ui ), nS(ui ) max mS ( ui ), nS ( ui ) S ( ui ) S ( ui ) It implies that min mS (ui ), nS (ui ) e (u ) (u ) S i S i
1 min m ( u ), n ( u ) 12 max mS (ui ), nS (ui ) 2min mS (ui ), nS (ui ) S i S i S ( ui ) S ( ui ) max mS (ui ), nS (ui ) e 0. (u ) (u ) S i S i
Thus, we have concluded that mS (ui ), mS (ui ) nS (ui ), nS (ui ) , for all ui U . (EN3). It is clear that
S c ui , mS (ui ), mS (ui ) , nS (ui ), nS (ui ) : ui U . Therefore, from (18), we get
H 2 ( S ) H 2 ( S c ). (EN4). Suppose that mT (ui ) nT (ui ), mT (ui ) nT (ui ), for each ui U and S T , i. e.,
mS (ui ) mT (ui ), mS (ui ) mT (ui ), nS (ui ) nT (ui ), nS (ui ) nT (ui ), for each ui U . Then, it follows that mS (ui ) nS (ui ) and mS (ui ) nS (ui ). From (18), we have 12 mS (ui ) mS (ui ) 1 H 2 (S ) p i 1 S (ui ) S (ui ) p
1 n ( u ) n ( u ) 2 S i(u ) S i (u ) e S i S i
12 nS (ui ) nS (ui ) (u ) (u ) S i S i
1 m ( u ) m ( u ) 2 S (i u ) S (iu ) e S i S i .
Therefore, p
H 2 ( S ) 1p 12 (2 nS (ui ) nS (ui )) e
1 (2 m ( u ) m ( u )) S i S i 2
i 1
1 (2 n (u ) n (u )) 12 (2 mS (ui ) mS (ui )) e 2 S i S i and H 2 (T )
p
1 p
i 1
1 2
( 2 n T ( u i ) n T ( u i )) e
12 ( 2 m T ( u i ) m T ( u i )) e
1 2
1 2
( 2 m T ( u i ) m T ( u i ))
( 2 n T ( u i ) n T ( u i ))
.
Since S T , we have nS (ui ) nS (ui ) nT (ui ) nT (ui ) and mS (ui ) mS (ui ) mT (ui ) mT (ui ). Therefore, H 2 ( S ) H 2 (T ). Similarly, if mT (ui ) nT (ui ), mT (ui ) nT (ui ), for each ui U and S T , then we can easily prove that H 2 ( S ) H 2 (T ).
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Highlights
1. New divergence and entropy measures are introduced for IVIFSs and compared with existing measures. 2. An interval-valued intuitionistic fuzzy COPRAS is developed. 3. The entropy and Shapley function based procedure for criteria weights is constructed. 4. The proposed approach is applied for an HWR facility selection problem. 5. Comparative work and sensitivity analysis from the case study are given.
S0360-8352(19)30609-6 https://doi.org/10.1016/j.cie.2019.106140 CAIE 106140
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Please cite this article as: Raj Mishra, A., Rani, P., Mardani, A., Raj Pardasani, K., Govindan, K., Alrasheedi, M., Healthcare Evaluation in Hazardous Waste Recycling Using Novel Interval-Valued Intuitionistic Fuzzy Information Based on Complex Proportional Assessment Method, Computers & Industrial Engineering (2019), doi: https://doi.org/10.1016/j.cie.2019.106140
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Healthcare Evaluation in Hazardous Waste Recycling Using Novel Interval-Valued Intuitionistic Fuzzy Information Based on Complex Proportional Assessment Method Arunodaya Raj Mishra1, Pratibha Rani2, Abbas Mardani3, Kamal Raj Pardasani4, Kannan Govindan5, Melfi Alrasheedi6 1Department
2Department
3Department
4 Department
5
of Mathematics, Govt College Jaitwara, Satna-485221, M P, India, email: [email protected]
of Mathematics, National Institute of Technology, Warangal, Telangana-506004, India, email:[email protected]
of Marketing, Muma Business of School, University of South Florida, Tampa, FL 33813, United States, email: [email protected]
of Mathematics, Bioinformatics and Computer Applications, MANIT, Bhopal-462051, M P, India, email: [email protected]
Center for Engineering Operations Management, Department of Technology and Innovation, University of Southern Denmark, Denmark DK-5230 Odense M, Denmark, email: [email protected]
6 Department
of Quantitative Methods, School of Business, King Faisal University, Saudi Arabia, email: [email protected]
Abstract Nowadays, the selection of the appropriate safety and health evaluation facility in hazardous waste recycling organizations has become a challenging issue, mainly in developing countries. Evaluation of hazardous waste recycling facility selection can be considered as a complex multicriteria decision-making problem that contains multiple-options solutions with conflicting tangible and intangible criteria. This paper is proposed a novel method based on the Complex Proportional Assessment (COPRAS) with multi-criteria decision making for interval-valued intuitionistic fuzzy sets. The proposed method is derived from aggregation operators of interval-valued intuitionistic fuzzy sets; in addition, a few amendments in the conventional complex proportional assessment approach and a process for computation of criteria weights are presented. For calculating the weights of the criteria the interval-valued intuitionistic fuzzy information measures are proposed. The present approach mainly addresses the correlative multi-criteria decision-making problems; therefore, the criteria weights are calculated in the form of Shapley values. Recently, increasing challenges for environmental issues require the development of hazardous waste recycling facility selection with respect to different criteria; therefore, the practical problem of hazardous waste recycling facility selection is presented, which proves the validity and feasibility of the proposed approach. Finally, a comparison is made with existing approaches to validate the developed method. Keywords: Information measures; Safety and health; Hazardous waste; Interval-valued intuitionistic fuzzy sets; Complex proportional assessment; Multi-criteria decision making.
1. Introduction A variety of wastes and sub-products are generated by various industrial sectors, such as food, healthcare, and mining (Guimarães et al., 2018). Roughly 40% of the wastes generated by these sources are classified as hazardous, which must be well treated in a way to prevent from appearing negative impacts on the environment and people’s health (Guimarães et al., 2018). In many industries like healthcare, it is important to avoid hazards, and a healthy environment is necessary for delivering safe outcomes. As a result, risk assessment is an important activity for many organizations through which they attempt to explore the methods for the mitigation or avoidance of hazards and risks that may arise within their site (Nahrgang et al., 2011). Remember that the majority of biological hazards come fromthe facilities used for waste treatments or the place wherein people are handling manually the untreated waste. During the waste treatment process, workers might be in direct or indirect contact with body fluids and blood that may leak out of used containers. In this procedure, they may also be in contact with airborne pathogens. The assessment of health and safety conditions in the work place with risky facilities of waste recycling is a complex MCDM problem that involves multiple and often contradictory qualitative and quantitative criteria (Guo et al., 2018; Hu et al., 2019; Tsai C. Kuo, 2013; Tremblay & Badri, 2018). Cost and benefit analysis is the general model for the analysis of social and private decisions. The cost and benefit analysis identify, measure and aggregate the costs and benefits to rank and select the optimal alternatives (Dompere, 1995). There are different subjectivity, vagueness and imprecision characteristics that should be accounted in different perspectives in social costing (Rothenberg, 1969). The concept of cost-benefit analysis is typically used to assess the environmental safety and health evaluation (Abrahamsen et al., 2018; Riaño-Casallas & Tompa, 2018). In addition, an economic index like the cost-benefit ratio is employed to compare various available decision alternatives. Such an index can also show uncertainty through the incorporation of the expected cost offailure. The extensive use of the risk-cost-benefit formula is due to its straightforwardness and simple interpretation of the obtained outcomes in monetary terms. Khadam and Kaluarachehi (2004) discussed in detail the limitations of the risk-cost-benefit approaches that are currently applied to assessing the environmental safety and health status. In the theory of fuzzy sets (FSs), intuitionistic fuzzy sets (IFSs), and interval-valued intuitionistic fuzzy sets (IVIFSs), the information (divergence and entropy) measures play
aimportant role in the study of uncertainty (Arya & Yadav, 2018; Çalı & Balaman, 2019; Sirbiladze et al., 2018; Zhang et al., 2019). At first, Zadeh (1969) studied the notion of entropy measure in FSs. Confirming the probability entropy, De Luca and Termini (1972) firstly presented the axiomatic definition of fuzzy entropy. Pal and Pal (1989) defined an exponential type entropy measure for FSs. Next, Burillo and Bustince (1996) originated the idea of IF-entropy and IVIFentropy. Furthermore, Szmidt and Kacprzyk (2001) pioneered IF-entropy with their geometrical interpretation. Hung and Yang (2006) developed the IF-entropy using the axioms of probability. Afterward, different researchers would analyze the concept of entropy in FSs and IFSs (Mishra & Rani, 2019; Narayanamoorthy et al., 2019; Qi et al., 2015; Zhang et al., 2019). Divergence measure is an essential procedure to estimate the distinction between elements. In FSs philosophy, Bhandari and Pal (1993) pioneered the notion of divergence measure. Similar to FSs, Vlachos and Sergiadis (2007) established the idea of IF-divergence measure and applied it to the image segmentation, medical problems, and pattern recognition. Next, Montes et al. (2015) presented a divergence measure for IFSs with new definitions and properties. Mishra et al. (2017) proposed new Jensen-exponential divergence measures; then, applied them to the development of a new MCDM technique with incomplete criteria under the IFSs environment. Mishra et al. (2019c) extended the divergence measure for hesitant fuzzy sets and utilized them to develop the WASPAS approach under hesitant fuzzy sets context. Further, Rani et al. (2019a) studied new divergence measure for Pythagorean fuzzy sets and applied to focus on the development of Pythagorean fuzzy VIKOR method. Currently, the concept of IVIF-divergence measure is extended, and until now, only a few researchers have focused onIVIF-divergence measure (Mishra, Chandel, et al., 2018; Mishra & Rani, 2018; Mishra, Singh, et al., 2019; Pratibha Rani et al., 2019). Based on compromise programming, one of the new methods called Complex Proportional Assessment (COPRAS), originated by Zavadskas et al. (1994), is a sensible and well-organized approach to processingthe information. The COPRAS approach has some key features as follow: It provides a valuable and suitable way to find a solution tothe MCDM problems (Anthony et al., 2019; Dey et al., 2017; Garg & Nancy, 2019; Roy et al., 2019; Wang et al., 2016). It assumes both aspects of criteria using Complex Proportional Evaluation, which provides moreexact information compared to different approaches proposed basically for solving the benefit or cost criteria. Finally, it describes the ratios to the optimal and the worst solutions at the same time.
Recently, due to the increasing complexity of MCDM problems, some authors extended the classical COPRAS method within the context of FSs, IVFSs, IT2FSs, HFSs and hesitant fuzzy linguistic sets (Mukhametzyanov & Pamucar, 2018). For instance, Bekar et al. (2016) introduced a new COPRAS-based technique with grey relations to evaluate the fuzzy MCDM Total Productive Maintenance model. Hajiagha et al. (2013) developed the COPRAS approach under an uncertain environment where the uncertainty is depicted in terms of IVIFSs. Keshavarz Ghorabaee et al. (2014) studied a new COPRAS-based approach to evaluate the supplier selection problem under Interval Type-2 fuzzy sets (IT2FSs). Vahdani et al. (2014) introduced the COPRAS method reflecting the subjective opinions and objective information of the real-life MCDM problems. Wang et al. (2016) proposed the COPRAS and Analytic Network Process (ANP)-based Failure Modes and Effects Analysis (FMEA) model for assessing and grading the risk of failure modes within the IVIF context. Zheng et al. (2018) extended the COPRAS approach with HLTSsto evaluate the severity of the chronic obstructive pulmonary disease. Mishra, et al. (2019) introduced an analytical COPRAS approach to explain the MCDM problems under HFSs. As the generalization of IFSs, IVIFSs are the vast growing environment of research, and these methodsare more flexible intackling the uncertainty and imprecision that may occur in daily life problems. Due to its capability, the present work focuses on the IVIF environment. Entropy and divergence measures, which are two critical topics in the FS theory and IVIFS theory, are widely implemented to different areas of decision making, medical diagnosis, and image segmentation. Thus, it is exciting to study the entropy and divergence measures of IVIFSs. As the exponential function has an advantage over the existing ones, this paper proposes the new IVIF-entropy and IVIF-divergence measures. However, a review of the literature shows that the entropy and divergence measures-based COPRAS approach has not been proposed under the IVIF environment, where the characteristics among the criteria are interdependent.Therefore, this paper proposes the IVIF-COPRAS method to handlethe problem of healthcare evaluation in hazardous waste recycling. The most significant contributions of this study are presented as follow: (i)
The novel entropy and divergence measures for IVIFSs are introduced and compared with existing ones.
(ii) Recently, various MCDM methods have been developed, in which the criteria have interrelated characteristics. However, these conjectures are despoiled in various cases. In this study, to cope with interdependent characteristics among the criteria, we developed Shapley values to evaluate the significance of criteria. (iii) The classical COPRAS method is extended with IVIFSs, in which the Shapley values are considered as the criteria weights. (iv) The developed method is applied to hazardous waste recycling facility selection and the performance of the proposed approach is compared with various existing ones. The paper is structured as follows: Section 2 provides the literature review and application decision making for evaluation of safety and healthy problems in hazardous waste recycling. Section 3 shows the fundamental concepts related to IVIFSs and information measures. Section 4 presents new IVIF-divergence and IVIF-entropy measures with counter-intuitive cases of existing measures. Section 5 proposes the COPRAS method based on Shapley function and entropy and divergence measures under the IVIF-environment. Also, a decision-making problem of HWR facility selection with IVIFSs is presented. Further, a comparision and sensitivity analysis are discussed to enlighten the applicability and feasibility of the proposed approach. Finally, Section 6 presents the conclusions of the whole study. 2. Evaluation of safety and health status in hazardous waste recycling People collect wastes and work with waste recycling systems may be subjected to a variety of risks and infections that, in many cases, may remain untreated. This may occur because of a limitation in having access to healthcare facilities. Therefore, to enhance the job-related safety of those who collect and recycle wastes is a crucial step towards the improvement of their social welfare. For this purpose, in the first step, there is a need for finding the actual job-related risks that are generally accompanied by any activity related to managing and working with solid wastes (Bleck & Wettberg, 2012). This is because of the fact that numerous toxic substances are possible to be given off in the course of recycling and disposing, which can simply lead to negative impacts on the environment, hence being a threat to people’s health status (Zheng et al., 2013). Such toxic substances are of a high risk not only to human beings but also to the whole environment and ecosystem (Cao et al., 2016; Long et al., 2013) through a variety of exposure ways (Song & Li, 2015; Zeng et al., 2016). Generally, workers are subjected to such negative impacts by being
exposed directly through their skin and inhalation system, whereas other people may be exposed through dust, smoke, food, and drinking water contamination (Robinson, 2009). Waste comes from a variety of sources, including the operations done for pollution control or decontamination processes, substances from contaminated soil remediation activities, adulterated materials, and activities done to purify the raw materials. The list can be also extended toalkaline or acidic substances, oils, sludge, plastics, wood, ashes, slag, metal, fibers, paper, glass, and rubber (Nakomcic-Smaragdakis et al., 2016). The research into this issue was carried out in a developed country, while the condition of developing countries has remained unknown, where the hazard can be more significant. Heavy metals, for instance, cadmium, chromium lead, zinc, and copper are some of the significantly-dangerous substances that can comeout of the waste recycling systems. They can be hardly removed from nature; they can be accumulated in human beings’ vital organs. The developed countries have made it a legal obligation to recover materials from waste flows in a formal way (Favot et al., 2016; Morris & Metternicht, 2016; Zhou et al., 2017). In these countries, strict regulations have been provided for waste recycling processes aiming at reducing the negative environmental impacts (Cesaro et al., 2018). On the contrary, developing countrieshave various informal recycling methods (Ardi & Leisten, 2016; Salhofer et al., 2016) such as mechanical operations, burning waste in open spaces, and chemical leaching processes, which are often carried out with no formal control. A piece of waste is hazardous in case it has any one of the following features: toxicity, reactivity, corrosiveness, or ignitability. Materials possessing such characteristics bring about the immediate or long-lasting risk to human beings, plants, animals, and the whole environment. Two factors that have a significant contribution to the stable increase of hazardous wastes are the growth of world population and the rise of industrialization. Nowadays, the government, the general public, and industrial sectors are aware of the pressing need for dealing effectively with the problem of hazardous wastes (Guo et al., 2018; Hu et al., 2019). Among the most extensively used methods for wastes disposition are disposition, incineration, and landfilling. On the other hand, burning and deposition of hazardous waste can, in turn, lead to an intense strain upon the environment, which can be because of the potential pollution of the water resources and the carbon dioxide that can be released from incineration sites. Hazardous wasteis given off into environment threats people's health through contaminated water and air. Recycling is known as one of the efficient ways that can manage hazardous wastes (Orloff & Falk, 2003). HWR facilities are capable
of treating, storing, or disposing of such wastes. If potential risks to safety and health are effectively prevented, workers will be able to work with the HWR facilities with a minimum level of risk. According to Cloquell-Ballester et al. (2007), the current decision-making approaches applied to the management of hazardous wastes generally have two disadvantages: the compensation problem and the problem of identifying an “exact boundary” between sub-ranges. They introduced an Elimination Et Choix Traduisant la REalité (ELECTRE) method to evaluate the importance of environmental effects based on comparative and sensitivity analyses. In another study, an MCDM model was employed by Rousis et al. (2008) to assess and rank 12 electrical waste management systems used in Cyprus. Their findings showed that the best system applicable to the Cyprus context is to partially disassemble and send the recyclable substance to the native market and disposethe residues at landfill spots. A multi-criteria mixed-integer linear programming model was introduced by Erkut et al. (2008) as a solution to the location-allocation problem of managing solid waste at a regional level. They made a comparison between the regional and prefectural planning for managing the solid wastes in Central Macedonia. On the basis of fuzzy logic, Peche and Rodríguez (2009) suggested an approach for the assessment of the environmental impacts coming from performing the projects and activities in cases wherethere is only limited information concerning the impacts. They applied fuzzy numbers to the environmental impact evaluation model to describe the impact properties. In addition, the MCDM technique was used by Dursun et al. (2011a) in order to perform an analysis based on the multi-level hierarchical structure and fuzzy logic in a way to effectively evaluate the alternatives to healthcare waste treatment. Taking into consideration the basis of fuzzy logic and multi-level hierarchy structure, Dursun et al. (2011b) attempted to evaluate the technologies of waste disposal. In addition, Liu et al. (2013) initiated a novel method called VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) to select the healthcare wastes (HCW) treatment. They validated their approach by applying it to solving the HCW selection problem in China. In another study, Liu et al. (2014) also used interval 2-tuple linguistic numbers (I2TLNs) using Optimization Method by Ratio Analysis (MOORA) plus full multiplicative form (MULTIMOORA) (Brauers & Zavadskas, 2010) method (ITL-MULTIMOORA) in a way to choose a proper technology for waste treatment in Shanghai, China. In addition, Voudrias (2016) provided wide-ranging details regarding the technologies of HCW treatment and also assessed
them employing analytic hierarchy process (AHP). Lee et al. (2016) carried out a study in England wherein AHP was employed as an instrument for exploration of the “optimal” technique that can be adopted for HCW treatment. Shi et al. (2017) introduced a hybrid MCDM method originating from the Multi-Attributive Border Approximation Area Comparison (MABAC) approach and used I2TLNs for the purpose of assessing and choosing the “best” method for HCW disposal. They also took into consideration the perception of numerous stakeholders. Xiao (2018), who introduced a D number theory-based approach, conducted one of the most recent investigations on the HCW treatment technologies. More recently, Liu et al. (2019) proposed an approach using Hamy-mean operators and intuitionistic uncertain linguistic variables for the evaluation of the best equipment of HCW treatment. Hinduja and Pandey (2018) developed a hybrid framework using the decisionmaking trial and evaluation laboratory (DEMATEL), ANP, and AHP for IFSs to predict and selectthe best HCW technique for Chhattisgarh, India. Hatami-Marbini et al. (2013) proposed anMCDM model, known as fuzzy-AHP-ELECTRE, based on an extended fuzzy method in the context of HWR. 3. Basic Concepts 3.1 Preliminaries As the FSs initiated by (Zadeh, 1965) are more capable to handle the uncertainty, these FSs have received extensive attention from scholars in the decision-making field of study. (Atanassov, 1986)) pioneered the notion of IFSs, which are distinguished by three parameters: belongingness degree, non-belongingness degree, and hesitation margins. As a result of ever-increasing intricacy of socio-economic surroundings, various researchers have extensively applied IFSs to MCDM problems (Mishra et al. 2017b; Stanujkic and Karabasevic 2018, Mishra and Rani 2019). In several cases, it is very complex to evaluate the exact degrees of these functions; thus, the degrees may be considered as intervals. Based on this idea, Atanassov and Gargov (1989) developed the notion of IVIFSs, which are expressed by intervals rather than the exact numbers and depicted by the interval belongingness, non-belongingness, and hesitation functions. Compared to IFSs, the IVIFSs are more prominent, dealing with uncertainty and imprecision because of their capability to handle the hesitant information. As a result, the utilization of IVIFSs has received more attention from several authors.
Recently, many researchers have focused their attention on the development of novel MCDM approaches under the IVIF context. To measure the uncertainty degree in IVIFSs, Liu et al. (2005) introduced the axiomatic conditions that an IVIF-entropy should comply with. Chen et al. (2010) developed an IVIF-entropy measure with its axiomatic properties and application in MCDM. Wei and Zhang (2015) defined IVIF-entropy and applied it to finding the criteria’s weights in MCDM problems. Rani et al. (2018) introduced IVIF-entropy and IVIF-similarity measures with their application in MCDM. Mishra, et al. (2018) introduced the bi-parametric IVIF-entropy and IVIFsimilarity measures and then utilized them to develop a Portuguese acronym for Interactive MultiCriteria Decision Making (TODIM) technique under IVIF context. For instance, Zhang et al. (2010) studied an IVIF-divergence measure and then utilized it to solve the pattern recognition problems. Ye (2011) developed a fresh MCDM method with IVIF-divergence measures. In another study, Meng and Tang (2013) presented an IVIF-divergence measure, which is used to assess the criteria and experts’ weights in the interactive MCDM method. (Meng & Chen, 2015) developed an MCDM approach based on novel IVIF-divergence measure and Shapley mapping associated with 2-additive measures. In addition, an IVIF-entropy and IVIF-divergence measures were developed by Mishra and Rani (2018), and they were used to discuss the application of weighted aggregated sum product assessment (WASPAS) approach. In MCDM approaches, evaluation of decision experts (DEs),weight vectors, and aggregation of information for options are the significant aspects for DEs. Literature also consists of many approaches proposed to evaluate the criteria weights and aggregation of the information under different fuzzy atmospheres. To compute the criteria weights, the past studies mentioned that diverse methods are based on the presumption that all the members of the set are independent and criteria have an equal significance. Thus, to handle the correlative MCDM problems, Sugeno (1974) originated the fuzzy measure that is a prominent device to model the interaction features among the members. For correlating the MCDM problem, the Shapley values are employed to quantify the importance of each criterion in all coalitions with identical probability (Shapley, 1953). In this study, an MCDM method is proposed under IVIFSs with interdependent interactions among the criteria. Here, various notions associated with IVIFSs and information measures are discussed.
Definition 3.1 (Atanassov and Gargov (1989): An IVIFS S on discourse set U u1 , u2 ,..., u p
is defined by
S ui , mS (ui ), nS (ui ) : ui U ,
(1)
where mS , nS :U 0,1 such that sup(mS (ui )) sup(nS (ui )) 1. The intervals mS (ui ) and
nS (ui ) represent the belongingness and non-belongingness degrees of an object ui in U , respectively. To make it easier, consider mS (ui ) mS (ui ), mS (ui ) and nS (ui ) nS (ui ), nS (ui ) , then
(2)
S ui , mS (ui ), mS (ui ) , nS (ui ), nS (ui ) : ui U , where 0 m S ( u i ) n S ( u i ) 1, m S ( u i ) 0
nS (ui ) 0.
and
The
interval
S (ui ) S (ui ), S (ui ) 1 ms (ui ) nS (ui ), 1 mS (ui ) nS (ui ) represents the hesitancy degree of u i to S . For a given ui U , the pair mS (ui ), nS (ui ) is said to be IVIFN (Xu, 2007). For simplicity, an IVIFN is expressed as S , , , , where , 0,1 , , 0,1 and 1. Definition 3.2 (Atanassov and Gargov (1989): Let S , T IVIFSs U , then different operations on IVIFNs are given by (a) S T iff mS (ui ) mT ( vi ), mS (ui ) mT (ui ), nS (ui ) nT (ui ) and nS (ui ) nT (ui ) for each u i U ;
(b) S T iff S T and S T ;
c (c) S ui , nS (ui ), nS (ui ) , mS (ui ), mS (ui ) : ui U ;
ui , mS (ui ) mT (ui ), mS (ui ) mT (ui ) , (d) S T : ui U ; nS (ui ) nT (ui ), nS (ui ) nT (ui )
ui , mS (ui ) mT (ui ), mS (ui ) mT (ui ) , (e) S T : ui U . nS (ui ) nT (ui ), nS (ui ) nT (ui )
Definition 3.3 (Liu et al., 2005): A mapping H : IVIFS U 0,1 is an IVIF-entropy if it holds the axioms: (EN1). H ( S ) 0 iff S is a crisp set; (EN2). H ( S ) 1 iff mS (ui ), mS (ui ) nS (ui ), nS (ui ) , for all ui U ;
(EN3). H S H S c ; (EN4). H ( S ) H (T ) if S T when mT (ui ) nT (ui ) and mT (ui ) nT (ui ) for each ui U or T S when mT (ui ) nT (ui ) and mT (ui ) nT (ui ) for each ui U . Definition 3.4 (Montes et al. (2015): A real function I : IVIFSs U IVIFSs U is called IVIF-divergence measure if it holds the given axioms: (D1). I S , T I T , S ; (D2). I S , T 0 iff S T ; (D3). I S W , T W I S , T , W IVIFSs U ; (D4). I S W , T W I S , T , W IVIFSs U . Definition 3.5 (Xu, 2007): Let S , , , be an IVIFN and 0, then S 1 1 , 1 1 , , .
(3)
From Eq.(1), the definition suggested byZ. Xu (2007)is employed as follows: Consider a set S S1 , S 2 ,..., S of ' ' IVIFNs, where S k k , k , k , k , k 11 . Then,IVIF-weighted arithmetic operator is given by k k k k k Sk 1 1 k , 1 1 k , k , k . k 1 k 1 k 1 k 1 k 1
Definition 3.6 (Xu, 2007): Consider S , , , be IVIFN. Then
(4)
S
(5)
1 1 , S , 2 2
are the score and the accuracy degrees of S , respectively, such that S 1,1 and S 0,1 .
Next, Xu et al. (2015) analyzed the score and accuracy functions and developed a modified score function for IVIFNs as follows: Definition 3.7 (Xu et al., 2015): Let S , , , be IVIFN. Then
* S
(6)
1 S 1 , S 1 S , 2
are said to be the normalized score and uncertainty values, respectively. Evidently, * S 0,1 and S 0, 1 .
Let S1 1 , 1 , 1 , 1
and S 2 2 , 2 , 2 , 2
be two IVIFNs. Then, a system is
described the comparability between two IVIFNs according to * S and S as follows: If * S1 * S 2 , then S1 S 2 , If * S1 * S 2 , then (i) if S1 S 2 , then S1 S 2 ; (ii) if S1 S 2 , then S1 S2 . In recent decades, various studies are available in theliterature indicating the postulation thatthe criteria have equal importance, the features under the criteria are independent, and the criteria weights are obtained by utilizing additive measures. However, these postulations have failed to deal with interrelated features among the criteria. To establish the interactions under interrelated criteria, Sugeno (1974) developed the idea of fuzzy measure (f-measure) as follows: Definition 3.8 (Sugeno, 1974): A map : P U [0,1] is said to beaf-measure on U if it holds the axioms:
(i) 0, U 1; (ii) If G T , then G T , G, T U . In MCDM problems, the term G is represented as the importance of criterion set G. The measure becomes an additive measure when G j , for any G P U . j G
In the correlated MCDM problems, the importance of each criterion is not only decided by itselfbut also gets the influence ofother criteria. To describe the overall interaction amongst the criteria, the Shapley function has been studied by many researchers. Definition 3.9 (Shapley, 1953): The Shapley function is an essential interaction index, which is defined by
j ,U
K U \ u j
n k 1! k ! n!
K u j K , u j U ,
(7)
where is a f-measure and K is distinguishedas a coalition shaped by the players in game theory, and it is observedas an association of criteria in the decision-making process. The Shapley value j is construedas a set of average importance of the concern of an object u j in any coalitions K \ u j . Furthermore, if there is no interaction, then
j ,U u j .
Property 3.1: If : P(U ) [0,1] is fuzzy measure, then j , U 0, u j U ; j 1(1) q. Property 3.2: If : P(U ) [0,1] is a fuzzy measure, then
q
,U U 1. j 1
j
Thus, j , U : j 1(1)q is a criterion weight vector.
3.2 Existing IVIF-Divergence Measures To quantify the degree of dissimilarity between IVIFSs, a variety of existing divergence measures from literature have been taken into consideration. According to Zhang et al. (2010):
p I Z JL ( S , T ) mS (ui ) ln i 1
1 mS (ui ) ln
(8)
mS (ui ) 1 2 mS (ui ) mT (ui )
1 mS (ui ) , 1 2 2 mS (ui ) mT (ui )
where mS (ui ) and mT (ui ) represent the average membership degree of ui to S and T , respectively, such that
mS (ui )
(9)
nS (ui ) nS (ui ) 1 mS (ui ) mS (ui ) 1 2 2 2 mS (ui ) mS (ui ) 2 nS (ui ) nS (ui ) , 4
nT (ui ) nT (ui ) 1 mT (ui ) mT (ui ) mT (ui ) 1 2 2 2
mT (ui ) mT (ui ) 2 nT (ui ) nT (ui ) . 4
Ye (2011) proposes Eq. (10) as follows: (10)
p m (u ) mS (ui ) 2 nS (ui ) nS (ui ) IY ( S , T ) S i 4 i 1
log 2
mS (ui ) mS (ui ) 2 nS (ui ) nS (ui )
1 2 mS (ui ) mS (ui ) 2 nS (ui ) nS (ui ) mT (ui ) mT (ui ) 2 nT (ui ) nT (ui )
p n (u ) nS (ui ) 2 mS (ui ) mS (ui ) S i 4 i 1
log 2
. 1 2 nS (ui ) nS (ui ) 2 mS (ui ) mS (ui ) nT (ui ) nT (ui ) 2 mT (ui ) mT (ui ) nS (ui ) nS (ui ) 2 mS (ui ) mS (ui )
Meng et al. (2013) suggests the following:
1 m (2um)S(nui )(u ) mS (ui ) 1 p S i S i log 2 I MT ( S , T ) mS ( ui ) p i 1 mS (ui ) nS (ui ) 1 m (u ) n (u ) m (umT) (uni ) (u ) S i S i T i T i
(11)
S
n (ui ) log 2 mS (ui ) nS (ui ) 1
, nS ( ui ) nT ( ui ) mS ( ui ) nS ( ui ) mT ( ui ) nT ( ui ) 1 m (2unS) (uni )(u ) S
i
i
S
where mP (ui ) mP (ui ) mP (ui ) and nP (ui ) nP (ui ) nP (ui ) for P S , T . And Meng et al. (2015) proposes Eq. (12) as follows: (12)
m (u ) 1 mS (ui ) I MC ( S , T ) S i log 2 2 1 12 mS (ui ) mT (ui ) i 1 p
2 mS (ui ) 3 mS (ui ) log 2 2 1 12 4 mS (ui ) mT (ui )
3 mS (ui ) nS (ui ) 2 mS (ui ) nS (ui ) log 2 2 1 12 4 mS (ui ) mT (ui ) nS (ui ) nT (ui )
mS (ui ) nS (ui ) 3 mS (ui ) nS (ui ) log 2 2 1 12 mS (ui ) mT (ui ) nS (ui ) nT (ui )
2 nS (ui ) 3 nS (ui ) nS (ui ) 1 nS (ui ) log log 2 2 2 2 1 12 4 nS (ui ) nT (ui ) 1 12 nS (ui ) nT (ui )
,
where mP (ui ) mP (ui ) mP (ui ) and nP (ui ) nP (ui ) nP (ui ) for P S , T . Next, the given examples illustrate the limitations of the existing divergence measures: Example 3.1: Consider S , T IVIFSs U where S u i , 0.1, 0.3 , 0.1, 0.3 : u i U and T
u , 0.3, 0.4 , 0.3, 0.4 i
: u i U . Then,
we
I Z JL S , T 0, IY S , T 0 and
obtain
I MT S , T 0, i.e., the measures (8)-(11) do not satisfy the postulate (D2) of Definition 3.5. Remark 3.1: In general, let us consider
T u , m (u ), m (u ) , n (u ), n (u ) m (u ), m (u )
S ui , mS (ui ), mS (ui ) , nS (ui ), nS (ui ) mS (ui ), mS (ui ) : ui U , i
T
i
T
i
T
i
T
i
T
i
T
i
: ui U .
From Example 3.1, we obtain that, I ZJL ( S , T ) 0, IY (S , T ) 0 and I MT ( S , T ) 0, whenever S T , the measures (8)-(11) violate the postulate (D2) of Definition 3.5.
Example 3.2: Consider S , T IVIFSs U where S u i , 0.2, 0.2 , 0.5, 0.5 : u i U and T
u , 0.3, 0.3 , 0.4, 0.4 i
: u i U . Consequently, we get that I Z JL ( S , T ) 0, IY (S , T ) 0
and I MT ( S , T ) 0, but S T . It implies that the measures (8)-(11) do not fulfill the axiom (D2) of Definition 3.5. Remark 3.2: Normally, consider
T u , m (u ), m (u ) m (u ) , n (u ), n (u ) n (u )
S ui , mS (ui ), mS (ui ) mS (ui ) , nS (ui ), nS (ui ) nS (ui ) : ui U , i
T
i
T
i
T
i
T
i
T
i
T
i
: ui U ,
Then I Z JL ( S , T ) 0, IY (S , T ) 0 and I MT ( S , T ) 0, but S T . Thus, the axiom (D2) of Definition 3.5 is violated for the divergence measures (8)-(11). 4. IVIF-Information Measures In this section, some existing information measures are discussed firstly. Next, new IVIFdivergence and IVIF-entropy measures are proposed, which are applied to the computation of the criterion weight in MCDM problems. 4.1. IVIF-Divergence Measure Due to the increase in complexity of the real-life problems, divergence measure for IVIFSs has attracted more attention of researchers in the decision-making field of study. However, existing information measures for IVIFS may be invalid, which will be explained later. To cope with such invalidity, a new divergence measure of order
for IVIFSs is proposed.
For S , T IVIFSs (U ), based on Ansari et al. (2018), a new IVIF-divergence measure is given by
p mS (ui ) 1 exp 1 mS (ui ) ln I S || T 1 1 2 ( ) ( ) m u m u 1 i S i T i
(13)
mS (ui ) nS (ui ) m (ui ) ln n u ( ) ln 1 2 mS (ui ) mT (ui ) S i 1 2 nS (ui ) nT (ui ) S
nS (ui ) 1 . nS (ui ) ln 1 2 nS (ui ) nT (ui )
Since 0 I S || T
1 1
1 41
1 . For simplicity, we normalize (13) and the cross-entropy
for IVIFSs is given by p mS (ui ) exp 1 ( ) ln I S || T m u S i 1 2 mS (ui ) mT (ui ) p 4 1 1 i 1 1
(14)
mS (ui ) nS (ui ) ( ) ln mS (ui ) ln n u 1 2 mS (ui ) mT (ui ) S i 1 2 nS (ui ) nT (ui ) nS (ui ) 1 . nS (ui ) ln 1 2 nS (ui ) nT (ui )
Now, the divergence measure for IVIFSs is given by
I S , T I S || T I T || S , p mS (ui ) 1 exp I S , T 1 m ( u ) ln S i 1 2 mS (ui ) mT (ui ) 2 p 4 1 1 i 1 mS (ui ) nS (ui ) ( ) ln n u mS (ui ) ln 1 2 mS (ui ) mT (ui ) S i 1 2 nS (ui ) nT (ui )
p nS (ui ) mT (ui ) exp 1 mT (ui ) ln nS (ui ) ln 1 2 nS (ui ) nT (ui ) 1 2 m ( u ) m ( u ) 1 i S i T i
(15)
mT (ui ) nT (ui ) m (ui ) ln u n ln ) ( 1 2 mS (ui ) mT (ui ) T i 1 2 nS (ui ) nT (ui ) T
nT (ui ) 2 . nT (ui ) ln 1 2 nS (ui ) nT (ui )
Theorem 4.1: Let S , T , W IVIFSs U and 1( 0). Then, I S , T fulfills the following postulates: (D1). I ( S , T ) I (T , S ); (D2). I ( S , T ) 0 iff S T ; (D3). I S , T I S c , T c and I S c , T I S , T c ; (D4). 0 I S , T 1; (D5). If S T W , then I S , T I S , W , I T , W I S , W . (D6). I ( S W , T W ) I ( S , T ), W IVIFSs U ; (D7). I ( S W , T W ) I ( S , T ), W IVIFSs U ; Proof: The proof is depicted in the Appendix. Remark 3.3: If IVIFSs transform to IFSs, then the divergence measure given by (13) will be converted to IF-divergence measure given by Ansari et al. (2018). The cross-entropy (14) is utilized to develop an entropy measure for IVIFSs. The above-mentioned examples 3.1 and 3.2 prove the advantage of the measure (15) over the others. The proposed measure (15) can be applied tothedecision-making process, edge detection, pattern recognition, etc. 4.2. IVIF-Entropy Measure Based on Vlachos and Sergiadis (2007), a relation between IVIF-divergence measure and IVIFentropy is developed. With the help of (15), we obtain
I S , S c 1
p mS (ui ) exp 1 m (u ) ln S i 1 2 mS (ui ) m c (ui ) p 4 1 1 i 1 S 1
mS (ui ) mS (ui ) ln 1 2 mS (ui ) mS c (ui )
nS (ui ) n (ui ) ln 1 2 nS (ui ) n c (ui ) S S
nS (ui ) nS (ui ) ln 1 2 nS (ui ) nS c (ui )
1 1.
It implies I S , S c 1
p mS (ui ) exp 1 m (u ) ln S i p 4 1 1 m u n u 1 2 ( ) ( ) i 1 S i S i 1
mS (ui ) nS (ui ) ( ) ln mS (ui ) ln n u 1 2 mS (ui ) nS (ui ) S i 1 2 nS (ui ) mS (ui ) nS (ui ) 1 u u nS (ui ) ln 1, ( ) ln 2 ( ) ln 2 4 1 S i S i 1 2 nS (ui ) mS (ui )
1
p 41
p mS (ui ) exp 1 m (u ) ln S i 1 m u n u 1 2 ( ) ( ) i 1 S i S i
mS (ui ) nS (ui ) ( ) ln mS (ui ) ln n u 1 2 mS (ui ) nS (ui ) S i 1 2 nS (ui ) mS (ui ) nS (ui ) 1 . nS (ui ) ln u u ( ) ln 2 ( ) ln 2 S i 1 2 nS (ui ) mS (ui ) S i
Thus, a relation between cross-entropy and entropy measures is obtained as follows:
H1 S 1 I S , S c , where
(16)
H1 S
1
p 41
p mS (ui ) exp 1 m (u ) ln S i 1 2 ( ) ( ) m u n u 1 1 i S i S i
(17)
mS (ui ) nS (ui ) m (ui ) ln n u ( ) ln 1 2 mS (ui ) nS (ui ) S i 1 2 nS (ui ) mS (ui ) S
nS (ui ) 1 . ( ) ln 2 ( ) ln 2 u u nS (ui ) ln S i 1 2 nS (ui ) mS (ui ) S i
Theorem 4.2: The measure H1 S , given by (17), is an IVIF-entropy. Proof: The proof is depicted in the Appendix. For each S IVIFS (U ), an IVIF-entropy is represented by H 2 S and given by
H 2 S
p
1 p
i 1
1 2 m in m S ( u i ), n S ( u i ) 1 m in m S ( u i ), n S ( u i ) S ( u i ) S ( u i )
2 m ax mS ( ui ), nS ( ui ) S i S i e S ( u i ) S ( u i ) 1 m ax m ( u ), n ( u )
12 max m S (u i ), n S (u i ) 2min mS ( ui ), nS ( ui ) S i S i S ( u i ) S ( u i ) max m S (u i ), n S (u i ) e . (u ) (u ) S i S i
1 min m ( u ), n ( u )
(18)
Theorem 4.3: The measure H 2 S , given by (18), is an IVIF-entropy. Proof: The proof is depicted in the Appendix. 4.3. Mathematical Comparison This section presents a comparison that illustrates the validity of proposed IVIF-entropy. Let S IVIFS (U ). For a real number p 0, De et al. (2000) introduced the set S p IVIFS (U ), which is given as follows:
(19)
p p p p S p ui , mR (ui ) , mR (ui ) , 1 1 nS (ui ) ,1 1 nS (ui ) : ui U .
Guo and Song (2014): H GS S
1 0.5 S ( ui ) S ( ui ) 1 p 1 1 m ( u ) n ( u ) m ( u ) n ( u ) , R i R i S i S i 2 p i 1 2
(20)
,
(21)
Wei and Zhang (2015):
H WZ
1 p mS ( ui ) nS ( ui ) mS ( ui ) nS ( ui ) S cos 2 2 S ( ui ) S ( ui ) p i 1
Meng and Chen (2015): 1 p min mS (ui ), nS (ui ) min mS (ui ), nS (ui ) , H MC S p i 1 max mS (ui ), nS (ui ) max mS (ui ), nS (ui )
(22)
Rani et al. (2018):
1 H R ( S ) p e 1 i 1 p
1 4
n
1 4
m
S
( u i ) m S ( u i ) 2
n S ( u i ) n S ( u i )
( u i ) n S ( u i ) 2 e m S (u i ) m S (u i ) S
1 4
m
S
( u i ) m S ( u i )
e
1 4
2
n S ( u i ) n S ( u i )
n
S
( u i ) n S ( u i )
2
m S ( u i ) m S ( u i )
(23)
1 .
Consider that S IVIFS U is defined by
S 6, 0.1, 0.2 , 0.6, 0.7 , 7, 0.2, 0.3 , 0.5, 0.6 , 8, 0.6, 0.7 , 0.1, 0.2 ,
(24)
9, 0.8, 0.9 , 0.0, 0.1 , 10, 1.0,1.0 , 0.0, 0.0. Taking into account the categorization of linguistic variables, De et al. (2000), Hung and Yang (2006), Xia and Xu (2012), and Rashid et al. (2018) considered S as “LARGE” on U . Using (19), we acquire
S 1 2 measured as “More or less LARGE.” S 2 measured as “Very LARGE.” S 3 measuredas “Quite very LARGE.” S 4 measured as “Very very LARGE.”
At this juncture, we calculate the entropy measures (17), (18), and (20)-(23), then the comparison outcomes are portrayed in Table 1. It is distinguished from the mathematical logical procedure that, IVIF-entropy measures have the following preference order:
H S1 2 H S H S 2 H S 3 H S 4 .
(25)
Table 1: Different IVIF-entropy values
S1 2
S
S2
S3
S4
H GS S
0.3551
0.2260
0.2042
0.1984
0.1864
HWZ S
0.5280
0.5460
0.4736
0.4206
0.3910
H MC S
0.3440
0.1950
0.1777
0.1983
0.1449
HR S
0.5321
0.5605
0.4718
0.4169
0.3870
H10.2 S
0.5102
0.4817
0.3562
0.3054
0.2675
H10.4 S
0.5396
0.5183
0.3908
0.3336
0.2949
H10.6 S
0.5913
0.5768
0.4479
0.3843
0.3444
H2 S
0.4755
0.4572
0.3976
0.3653
0.3344
In Table 1, the entropy measures except HWZ S , H MC S , and H RJ S satisfied the requirements mentioned in (25). The behaviors of IVIF-entropy measures H GS S , H1 S , and
H 2 S are thus reasonably good from the perspective of ordered linguistic variables. Remark 4.4: From the above-mentioned example, we can see that the entropy measures (17) and (18) successfully hold the order pattern given in (25), which shows their reliability. In this
paper, a correlative MCDM method is discussed with incomplete criteria weights information using proposed entropy measures. 5. Proposed IVIF-COPRAS Method for MCDM Here, the Shapley function-based COPRAS approach is developed to tackle the correlative MCDM problems under the IVIF doctrine. 5.1. Proposed IVIF-COPRAS Method with Shapley Function The Complex Proportional Assessment (COPRAS) approach pioneered by Zavadskas et al (1994) is extremely useful for solving MCDM problems. The key outcomes of the COPRAS are as follow (Mishra et al., 2018; Zheng et al., 2018): first, it can simultaneously take the ratios to the optimal and the worst solutions. Second, outcomes can be achieved in a very less time as compared to the different existing MCDM methods. Finally, it is computationally straightforward and easily comprehensible. Based on the benefits, the COPRAS approach has been widely implemented in various decision-making enviroments. In the conventional COPRAS approcah, the assessment of alternatives over the criteria is distinguished as crisp numbers. Nonetheless, in diverse circumstances, crispness is not enough to handle real-life MCDM problems. As IVIFSs have been widely implemented to solve the uncertain MCDM problems, the conventional COPRAS method is generalized. The procedural steps of IVIF-COPRAS approach are depicted as follows (graphically drawn in Fig. 1): Step 1: Create an IVIF-decision matrix.
For an MCDM problem, generate a set of alternatives B B1 , B2 ,..., B p and classify the set of
criteria G G1 , G2 ,..., Gq . The decision expert presents his/her assessment values i j of the alternatives B i i 1(1) p with respect to criteria G j j 1(1) q in the form of linguistic variables (LVs). Next, we transform LVs into IVIFNs, which constructs an IVIF-decision matrix M ij
pq
, where ij ij , ij , ij , ij , for experts.
Step 2: Assess the DEs weights
Let k 1 , 2 ,..., be the weight values of DEs. These DEs weights are considered as T
linguistic values and expressed in IVIFNs. Let ek mk , mk , nk , nk be a rating of the k th DE in terms of IVIFNs. According to Boran et al. (2009), the DEs’ weight is evaluated by mk mk m k mk k k mk nk mk nk , k 1, 2,..., . k m m mk k k mk k k m n k 1 k k mk nk
Here, k 0 and
l
(26)
k 1.
k 1
Step 3: Find the aggregated IVIF (A-IVIF) decision matrix
Let M ij k
k
be the decision matrix of k th DE, where ij mijk , mijk , nijk , nijk ,
k 1, 2,..., is IVIFNs. Consequently, to combine all the individual matrices and generate a
combined decision matrix, an A-IVIF decision matrix is constructed. To do this, let zij p q where zij aij , bij , cij , dij , i 11 p, j 11 q be the A-IVIF decision matrix, where k 1 k z ij k and l
l l l l k zij 1 1 mijk ,1 1 mijk k , nijk k , nijk k . k 1 k 1 k 1 k 1
(27)
Step 4: Evaluate the criteria weights. To distinguish the importance of the criteria, we have to compute the criterion weight. For correlative characteristics among the criteria, the weight vector is computed as follows: Step 4.1: Compute the entropy value of alternative Bi associated with criteria G j . A linear programming model (LP-model) for desirable fuzzy measure associated with criteria
G is developed when the information associated with criteria weights is completely unknown:
q
(28)
p
min H zij j , G j 1 i 1
0, G 1, s. t. L M L, M G , L M ,
where j , G is the Shapley value of criteria G j j 1 1 q and H zij is an entropy value for the IVIFN zij . Whereas, if the information associated with the criteria weights is partially known, then the LPmodel is assembled as follows: q
(29)
p
min H z ij j , G j 1 i 1
0, G 1, s. t . G j Y j , j 1(1) q , L M , L , M G , L M ,
where H zij is the entropy for the IVIFN zij and Y j y j , y j is its range. Utilizing (7), estimate the Shapley values of each criterion. Therefore, the Shapley values are employed as the criteria weights for the correlative MCDM problem. Step 5: Determine the normalized aggregated IVIF decision matrix. Normalize
the
A-IVIF
decision
matrix
zij
mn
into
N ij
p q
,
where
ij ij , ij , ij , ij is IVIFN, using the formula
ij
ij
aij
12
2 2 p aij bij i 1
cij
2 2 p cij dij i 1
, ij
bij
12
2 2 p aij bij i 1
,
dij , , ij 12 1 2 i 11 p, j 11 q. 2 2 p cij dij i 1
(30)
(31)
Figure 1: Framework of the IVIF-COPRAS method for MCDM
Step 6: Sum the criteria values for benefit-type and cost-type. In the proposed approach, each option is illustrated with its additions of maximizing criterion,
i , which is assigned to benefit type and minimizing criterion, i , which is assigned to cost type.
To evaluate the value of i and i , we implement the following procedure: s
i j , G ij , i 11 p,
(For benefit-type)
(32)
j 1 q
, G
i
j s 1
j
ij
, i 11 p.
(For cost-type)
(33)
In formulas (32) and (33), s describes the maximizing (benefit-type) criteria, q shows the entire criteria, and j , G is importance degree of the jth criteria. Step 7: Evaluate the relative degree of the alternative. The relative degree or weight i of option is evaluated by
i * i
(34)
p
min i * i *
i
i 1
p
* i
min * i
, i 11 p
i
* i
i 1
where * i and * i represent the score values of i and i , respectively. Next, we can also inscribe (34) as follows: p
i * i
i
i 1
p
i *
(35)
*
i 1
1
, i 11 p.
* i
Step 8: Obtain the preference order or degree. The preference degree of alternatives is evaluated according to the relative degree or weight of the alternative. The maximum relative weight determines the higher preference order (rank) of the alternative, which is the desirable one. Thus, B* max i , i 11 p. i
(36)
Step 9: Compute the utility degree. The utility degree i is computed by comparing the evaluated alternative with the optimal one. The range of the i lies from 0% to 100% representing the worst and the optimal ones, respectively. Therefore, the utility degree i of each alternative is calculated by
i
i 100%, i 11 p, max
(37)
where i and max are the relative degrees of the alternative. The proposed MCDM method allows us to evaluate the straight and comparative dependence of the relative degree and utility degree of alternatives associated with the criteria set.
5.2. Evaluation of HWR Facility Selection Problem using the IVIF-COPRAS method This section discusses an MCDM problem of selecting the HWR facility for the purpose of verifying the efficiency of the proposed IVIF-COPRAS approach. Here is presented the results obtained from a pilot study carried out for the Indian Health Service (IHS) in the Albuquerque region. It was aimed to assessthehealth and safety status in the case of chosen HWR facilities through the framework introduced in the present study. IHS formally provides American Indians and Alaska Natives with federal health services. Suchestablishment of health services for the members of tribes, which are federally-recognized,is actually originated from the relationships between thesetribes and the federal government. The Albuquerque area is broken out by administrativeregions of service units. The formal responsibility of IHS is improvingthe American Indians and Alaska Natives physically, mentally, socially, and spiritually to utmost levels. It attempts to provide healthy communities and healthcare systems of high-quality throughrobust partnerships and culturally-responsive practices. IHS is mainly aimed at ensuring that all-inclusive, culturally-proper personal, and public health services are completely available for American Indians and Alaska Natives; promoting a high quality of lifeby innovating the Indian health system into an optimally performing organization; and also strengtheningthemanagement and operations of IHS. Five alternative facilities within the Albuquerque area were chosen to take part in the present research through the IHS officials; they are Jicarilla Service Unit ( B1 ), Ute Mountain Ute Service Unit ( B2 ), Zuni Comprehensive Health Center ( B3 ), New Sunrise Regional Treatment Center ( B4 ), and Taos-Picuris Service Unit ( B5 ). One of the seasoned inspectors was chosen by the IHS officials to lead this effort. The lead inspector E1 invited a colleague from the Division of Facilities Planning Construction (DFPC) E2 and another colleague from the Division of Environmental Health Services (DEHS) E3 to join him in this collaborative initiative. The three
experts E1 , E2 , E3 of the evaluation team (i.e., the IHS inspector, the DFPC inspector, and the DEHS inspector)were chosen to make an evaluation on the HWR Facility option. In addition, the assessment team agreed to employ the linguistic variables listed in Tables 2 and 3 in the description of the important weights and the performance ratings of the criteria, respectively. The team took into account the following four criteria (which were established by IHS) for the purpose of assessing in asystematic way the potential safety and health hazards in waste recycling facilities all through the Albuquerque area: severity of occurrence ( G1 ), time of exposure ( G2 ), failure to detect the risk ( G3 ), and protective & preventive measures ( G4 ). Here, the criteria G1 , G2 , G3 are the benefit criteria and G4 is the cost criterion, and the scale of the criteria is specified by
0.4, 0.6 , 0.1, 0.3 , 0.15, 0.3 and 0.15, 0.45. Now,
the procedure for execution of IVIF-
COPRAS method is given by Step 1: Constructionof an IVIF-decision matrix. Each decision expert executes his/her opinions regarding the rating of HWR selection over the preferred criteria. Here, Table 2 represents the linguistic variables in the form of IVIFNs to estimate the relative importance of preferred assessment criteria. Table 3 expresses the preference of decision experts
E1 , E2 , E3
for the criterion weight. Based on the IVIFN scale, Table 4
presents the linguistic values for the performance of the HWR facility evaluated by decisionmakers. Table 5 expresses the individual decision opinions of each Ei concerning the performance of HWR selection. Table 2: Importance weights as linguistic terms Linguistic variables
IVIFNs
Very important (VS)
0.80, 0.95 , 0.0, 0.05
Important (S)
0.65, 0.75 , 0.15, 0.20
Modest (M)
0.45, 0.55 , 0.30, 0.45
Unimportant (IS)
0.20, 0.30 , 0.55, 0.70
Very unimportant (VIS)
0.00, 0.10 , 0.80, 0.90
Table 3: Importance weights of
Ei i 1, 2,3 as linguistic terms
E1
E2
E3
Very important
Modest
Important
IVIFNs
0.80,0.95 ,0.0,0.05
0.45,0.55 , 0.35,0.45
0.65,0.75 ,0.15,0.20
Weights
0.394
0.274
0.332
Linguistic values
Table 4: Significance rating of HWR facility option as linguistic values Linguistic terms
IVIFNs
Extremely good (XG)/ extremely high (XH)
0.90, 0.95 , 0.00, 0.05
Very very good (EEG)/ very very high (EEH)
0.80, 0.90 , 0.05, 0.10
Very good (EG)/ very high (EH)
0.70, 0.80 , 0.15, 0.20
Good (G)/ high (H)
0.60, 0.70 , 0.20, 0.25
Fairly good (FG)/ moderate high (OH)
0.50, 0.60 , 0.25, 0.30
Fair (F)/ moderate (O)
0.40, 0.50 , 0.35, 0.40
Fairly bad (FB)/ moderate low (OL)
0.30, 0.40 , 0.45, 0.50
Bad (B)/ low (L)
0.15, 0.25 , 0.55, 0.60
Very bad (EB)/ very low (EL)
0.10, 0.15 , 0.70, 0.75
Very very bad (EEB)/ very very low (EEL)
0.00, 0.10 , 0.85, 0.90
Table 5: Performance rating of HWR options from DEs as linguistic terms HWR alternative
B1
B2
B3
DEs
G1
G2
G3
G4
E1
OH
EEG
G
O
E2
H
FG
FB
OH
E3
OL
G
B
H
E1
EH
FB
FB
EEL
E2
OH
EEG
G
L
E3
O
EB
EB
EL
E1
O
G
EEG
EH
E2
OL
F
XG
O
E3
L
FB
EG
H
B4
B5
E1
H
FG
F
OL
E2
O
B
EG
OL
E3
OH
F
G
EL
E1
OH
XG
EG
EH
E2
H
EG
G
OH
E3
OL
EEG
EEG
H
Based on Tables 3-5 and Eq. (27), the A-IVIF-decision matrix is constructed and given in Table 6. Table 6: A-IVIF-decision matrix for HWR facility alternative
G1
G2
G3
G4
B1
0.406, 0.521 , 0.232, 0.330
0.614, 0.776 , 0.134, 0.204
0.398, 0.522 , 0.195, 0.418
0.517, 0.699 , 0.124, 0.208
B2
0.490, 0.732 , 0.098, 0.212
0.332, 0.778 , 0.122, 0.200
0.287, 0.403 , 0.375, 0.582
0.138, 0.217 , 0.440, 0.520
B3
0.267, 0.324 , 0.583, 0.664
0.4, 0.5 , 0.333, 0.420
0.707, 0.794 , 0.121, 0.201
0.552, 0.618 , 0.237, 0.359
B4
0.505, 0.614 , 0.131, 0.235
0.322, 0.435 , 0.226, 0.313
0.544, 0.758 , 0.128, 0.208
0.151, 0.201 , 0.677, 0.781
B5
0.481, 0.542 , 0.354, 0.427
0.830, 0.919 , 0.022, 0.072
0.621, 0.781 , 0.113, 0.197
0.561, 0.631 , 0.131, 0.261
Step 2: Computation of the criterion weight. To evaluate the criterion weight, (i)
The entropy measure (18) of each HWR facility Bi i 11 5 is computed associated with
criteria G j i 11 4 in Table 7. Table 7: Entropy values of A-IVIF-decision matrix for HWR facility
G1
G2
G3
G4
B1
0.8716
0.5728
0.8903
0.6683
B2
0.6588
0.7205
0.9006
0.8001
(ii)
B3
0.7350
0.9453
0.5006
0.7743
B4
0.7280
0.9303
0.6225
0.5373
B5
0.9052
0.2451
0.5585
0.6978
The LP- model (28) for the optimal fuzzy measure is constructed as follows:
min [0.1110{ (G1 ) (G 2 , G3 , G 4 )} 0.0506{ (G 2 ) (G1 , G3 , G 4 )} 0.0311{ (G3 ) (G1 , G 2 , G 4 )} 0.0293{ (G 4 ) (G1 , G 2 , G3 )} 0.0302{ (G1 , G 2 ) (G3 , G 4 )} 0.0399{ (G1 , G3 ) (G 2 , G 4 )} 0.0408{ (G1 , G 4 ) (G 2 , G3 )} 3.5657] (G1 ) (G1 , G2 ), (G1 ) (G1 , G3 ), (G1 ) (G1 , G4 ), (G2 ) (G1 , G2 ), (G2 ) (G2 , G3 ), (G2 ) (G2 , G4 ), (G3 ) (G1 , G3 ), (G3 ) (G2 , G3 ), (G ) (G , G ), (G ) (G , G ), (G ) (G , G ), (G ) (G , G ), 3 3 4 4 1 4 4 2 4 4 3 4 (G1 , G2 ) (G1 , G2 , G3 ), (G1 , G3 ) (G1 , G2 , G3 ), (G2 , G3 ) (G1 , G2 , G3 ), s. t. (G1 , G2 ) (G1 , G2 , G4 ), (G1 , G4 ) (G1 , G2 , G4 ), (G2 , G4 ) (G1 , G2 , G4 ), . (G , G ) (G , G , G ), (G , G ) (G , G , G ), (G , G ) (G , G , G ), 1 3 1 3 4 1 4 1 3 4 3 4 1 3 4 (G2 , G3 ) (G2 , G3 , G4 ), (G3 , G4 ) (G2 , G3 , G4 ), (G2 , G4 ) (G2 , G3 , G4 ), (G1 , G2 , G3 ) 1, (G1 , G3 , G4 ) 1, (G1 , G2 , G4 ) 1, (G2 , G3 , G4 ) 1, (G ) 0.4, 0.6 , (G ) 0.1, 0.3 , (G ) 0.15, 0.3 , (G ) 0.15, 0.45 . 1 2 3 4
(38)
Using MATHEMATICA, solve (38) to compute the fuzzy measure associated with criteria G thatis given by
(G1 ) 0.4, (G2 ) 0.3, (G3 ) 0.3, (G4 ) 0.4 (G1 , G2 ) (G1 , G3 ) (G1 , G4 ), (G2 , G3 ) 1 (G2 , G4 ), (G3 , G4 ) 0.4, (G1 , G3 , G4 ), (G1 , G2 , G3 ) 1 (G1 , G2 , G4 ) (G2 , G3 , G4 ) (G1 , G2 , G3 , G4 ). (iii) The
calculated
Shapley
values
are
GH , G 0.1167, GH , G 0.4834, 2
1
2
2
GH , G 0.1833 and GH , G 0.2166. 2
3
2
4
Step 3: Creation ofa Normalized A-IVIF-decision matrix. Using formulas (30) and (31), the normalized A-IVIF-decision matrix is computed and given in Table 8. Table 8: Normalized A-IVIF-decision matrix
G1
G2
G3
G4
B1
0.1166, 0.1497 , 0.0990, 0.1408
0.1443, 0.1824 , 0.0908, 0.1382
0.0960, 0.1259 , 0.1049, 0.2248
0.1695, 0.2291 , 0.0463, 0.0776
B2
0.1408, 0.2103 , 0.0418, 0.0904
0.0780, 0.1828 , 0.0867, 0.1355
0.0692, 0.0972 , 0.2017, 0.3130
0.0452, 0.0711 , 0.1642, 0.1941
B3
0.0767, 0.0931 , 0.2487, 0.2832
0.0940, 0.1175 , 0.2257, 0.2846
0.1706, 0.1916 , 0.0651, 0.1081
0.1809, 0.2026 , 0.0885, 0.1340
B4
0.1451, 0.1764 , 0.0559, 0.1002
0.0757, 0.1022 , 0.1531, 0.2121
0.1312, 0.1829 , 0.0688, 0.1119
0.0495, 0.0659 , 0.2527, 0.2915
B5
0.1382, 0.1557 , 0.1510, 0.1821
0.1951, 0.2160 , 0.0149, 0.0488
0.1498, 0.1884 , 0.0608, 0.1059
0.1839, 0.2068 , 0.0489, 0.0974
Here, the severity of occurrence ( G1 ), time of exposure ( G2 ), failure to detect the risk ( G3 ), and protective & preventive measures ( G4 ) are given criteria to evaluate HWR facility alternative selection problem. As mentioned earlier, the criteria G1 , G2 and G3 are the benefit criteria and G4 is the cost criterion. Using (32)-(37), the values of i , * i , i , * i , i and i of
Bi i 11 5 are calculated related to criteria G j j 1 1 4 , which are given inTable 8. Table 9: The computational outcome of IVIF-COPRAS approach for HWR facility selection Alternatives
i
* i
i
* i
i
i
B1
0.1026,0.1315 , 0.1584,0.2325
0.4608
0.0394, 0.0548 , 0.5140, 0.5748
0.2514
0.6094
83.15%
B2
0.0677,0.1340 , 0.1579,0.2323
0.4529
0.0100,0.0158 , 0.6762,0.7011
0.1621
0.6834
93.25%
B3
0.0873,0.1049 , 0.2509,0.3127
0.4071
0.0423,0.0479 , 0.5914,0.6470
0.2129
0.5826
79.49%
B4
0.0788,0.1058 , 0.1765,0.2418
0.4416
0.0109,0.0147 , 0.7423,0.7657
0.1294
0.7329
100%
B5
0.1410,0.1611 , 0.0628,0.1262
0.5283
0.0431,0.0489 , 0.5201,0.6038
0.2420
0.6827
93.15%
From Table 9, the preference order of HWR facility selection is B4 B2 B5 B1 B3 ,thus, B4 is the best HWR facility. 5.3. Sensitivity Analysis In this paper, a sensitivity analysis was done to investigate the proposed approach behavior. Four different criteria weight sets are taken and depicted in Table 10. In this table, for every set, one of the criterion has the maximum weight, while the other ones have lower weights. By applying this process, an ample scope of criteria weights was created to investigate the sensitivity ofthe developed approach to variation of the criteria weights.
Figure 2: Results of
i for each alternative with different values of weight sets
Table 10: Different criteria weight sets for HWR facility alternative Weight sets
G1
Set- I Set – II
G2
G3
G4
0.1167
0.4834
0.1833
0.2166
0.2166
0.1167
0.4834
0.1833
Set – III
0.1833
0.2166
0.1167
0.4834
Set – IV
0.4834
0.1833
0.2166
0.1167
The sensitivity analysis results described in Fig. 2 showed that the relative degree i couldchange over different criteria weight sets and the rank of HWR facility alternative. For example, when DEs give the weight set-I, III, and IV, the ranking of HWR facility ranking is
B4 B2 B5 B1 B3 , whereas when the evaluation weight set-II is taken, then HWR facility ranking becomes B4 B2 B5 B3 B1. From the above discussion, it is concluded that the HWR facility selection is dependent on and sensitive to these criteria weight sets. Therefore, the proposed approach has acceptable stability with different weight sets. 5.4. Comparison and Discussion In this section, we first compare the proposed IVIF-COPRAS approach with the IVIF-TODIM adopted from Mishra and Rani (2018) method in the same environment (an example is presented in Section 5.1). Furthermore, a comparison is made between the proposed and IVIF-TODIM approaches. In the same environment, the classical TODIM method is implemented, in which the evaluation values are given in the form of IVIFNs. In the following, the computation steps for the TODIM method are discussed. Step 1: Determining the relative weight vector. The relative weight vector jr ( , G ) of G j associated withcriterion Gr is calculated using the formula jr , G
j , G , where r , G max j , G and j , G is the criteria r , G
weight. Step 2: Calculating the dominance matrix. The degree of dominance of alternative Bi associated withan alternative Bt is computed using the expression
jr , G I ( zij , ztj ) if zij ztj , q jr , G j 1 j Bi , Bt if zij ztj , 0, q 1 jr , G I ( ztj , zij ) j 1 , if zij ztj jr , G
(39)
where the term represents the reduction factor that controls the impact of loss, I ( zij , ztj ) describes the degree of dissimilarity between IVIFNs zij and ztj , and jr , G shows the criterion Shapley value. Step 3: Evaluating the overall dominance degree. The overall dominance degree j Bi , Bt of alternative Bi over Bt is evaluated using the expression (40)
q
Bi , Bt j Bi , Bt , i, t 1(1) p. j 1
Step 4: Determining the overall prospect or global value. Based on the conventional TODIM method, we determine the overall dominance degree of alternative Bi associated with alternative Bt as follows:
B , B min B , B , i 1(1) p. B max B , B min B , B p
p
i
k 1
i
t
i
p
i
k 1
i
k 1
t
(41)
p
i
t
i
k 1
i
t
Step 5: Considering the preference degree of the options and choosing the optimal one. Preference order of the alternatives based on the increasing order of global values and choose the optimal one. The higher overall prospect value determines the optimal alternative. The results obtained by applying the IVIF-TODIM method to the HWR facility selection problem mentioned in Section 5.1 are given as below: Step 1: The relative weight vector is evaluated as 1r ( , G ) 0.2414, 2 r ( , G ) 1.0000,
3r ( , G ) 0.3792 and 4r (, G) 0.4481.
Step 2: Through the use of Eq. (39), the dominance degree of each HWR alternative
Bi i 11 5 is calculated under the criteria G j j 11 4 at 2.5; thus, the dominance matrix is expressed as follows: 0 0.2014 1 0.1953 0.1433 0.0809
0.1232 0 0.3125 0.0442 0.2818
0.3195 0.0876 0.0809 0.5110 0.0723 0.4610 0.2792 0.1649 , 0 0.4567 0 0.2164 0.2697 0.1323 0
0.0705 0.1426 0.1209 0 0.1828 0.1344 0.1073 0 2 0.3698 0.3487 0.0956 0 0 0.3136 0.2782 0.0368 0.1220 0.1578 0.2509 0.2812 0 0.2965 3 0.1059 0.0878 0.0583
0.0772 0 0.1782 0.1592 0.2055
0.3166 0.4092 0.9266 , 0.7294 0
0.4071 0.3373 0.2239 0.6850 0.6117 0.7896 0.2194 0.0125 , 0 0.8433 0.0632 0 0.0480 0.0165 0
0.1841 0.0102 0 0.8415 0.6760 0 4 0.0468 0.1479 0 0.2263 0.2014 0.8564 0.1012 0.1753 0.0410
0.2283 0.0221 0.0441 0.8010 0.1874 0.1873 . 0.9895 0 0 0.2165
Step 3: From Eq. (40), the overall dominance degree of each HWR alternative Bi i 11 5 is determined under the criteria G j j 11 4 , therefore, the overall dominance matrix is articulated as below: 0.2086 0.0652 0.0777 0.4375 0 1.1194 0 0.7156 0.3880 1.5388 0.5060 0.3351 0 0.0320 1.2663 . 0 0.7635 0.1438 0.3646 1.2062 0.1600 0.2568 0.5136 0.3819 0
Step 4: Through the use of Eq. (41), the overall prospect value Bi of each HWR facility option Bi is determined as follows:
B1 0.6938, B2 0.0000, B3 0.3097, B4 1.0000 and B5 0.3324. Step 5: Corresponding to Step 4, the ranking of the HWR is B4 B1 B5 B3 B2 ; thus, B4 is the best HWR alternative. According to the prospect theory, Kahneman and Tversky (2013) suggested that the factor of attenuation of losses is around 2.25, and [1, 2.5]. To observe the impact of the factor in the problem;various factor values were implemented from 1 to 2.5; then, the same ranking outcomes were achieved. Consequently, the HWR option B4 is the most desirable one for any value of . The outcome of IVIF-TODIM method not only determines the most desirable HWR alternative but also showsthe psychosomatic behaviors of the DEs. In this section, to illustrate the superiority of the IVIF-COPRAS approach, a comparison is also made between its results and those of the existing ones.In recent years, various decision-making approaches with different characteristics have been introduced for the safety and health evaluation in hazardous waste recycling context. Here, for the comparative study, we preferred the approaches that could be efficiently applied to the considered MCDM problem. Then, the approaches proposed by Hashemi et al. (2015), Kuo et al. (2015), and Tang (2017) were extracted from literature for the analysis purposes. To have an effective comparison, the Spearman's rank correlation rB between the developed approach and existing ones were calculated (see Table 12). As can be observed in Table 12, the correlation coefficients are greater than 0.6 (there is a high degree of dependency between ranking outcomes). As a result, the relationships between preference order outcomes are strong or very strong. Based on the research findings, the outcome of the developed approach was found consistent with the existing approaches, except the method of Kuo et al. (2015) and the IVIF-TODIM (Mishra and Rani (2018)). According to Tables 11 and 12, the methods developed by Hashemi et al. (2015), Kuo et al. (2015), and Tang (2017), as well as the IVIF-TODIM (Mishra and Rani (2018)) have rankings different from that of our proposed method (Except for best HWR facility, B4 ). The approach developed by Hashemi et al. (2015), which is based on ANP-Grey relational analysis (GRA), is easy to implement; however, it doesnot consider the simplification in the weight assessment
process, lacks a basis to ensure for feasible inconsistencies, and ignores the inherent interdependencies with the criteria. Kuo et al. (2015) developed a hybrid model integrating DANP and VIKOR methods, which provided a compromise solution, but it was unable to consider the bounded rationality of decision-makers. The method presented by Tang (2017) is based on the aggregation operator. Its drawback is that different aggregation operators typically provide different rankings. However, the proposed IVIF-COPRAS method is developed to avoid the shortcomings of existing methods and provide more reasonable results. Table 11: Comparative outcomes of ranking order with various approaches Approaches
Benchmark
Environment
Rank order
Optimal HWR Facility
Kuo et al. (2015)
DANP based VIKOR
Crisp Sets
B4 B5 B1 B3 B2
B4
FSs
B4 B1 B5 B2 B3
B4
methods Hashemi et al.
ANP and Improved GRA
(2015)
methods
Keshavarz
Extended WASPAS method
IT2FSs
B4 B5 B2 B1 B3
B4
Hesitant fuzzy Hamacher
HFSs
B4 B5 B1 B2 B3
B4
IVIFSs
B4 B1 B5 B3 B2
B4
IVIFSs
B4 B2 B5 B1 B3
B4
Ghorabaee et al. (2014) Tang (2017)
power weighted average (HFHPWA) Operator method Mishra and Rani
Divergence measure based
(2018)
TODIM method
Proposed IVIF-
Divergence measure based
COPRAS method
improved COPRAS method
Table 12: Ranking of the HWR facility with existing approaches and coefficient of correlation rB HWR facility
Kuo et al. (2015)
B1
Hashemi et al. (2015) 2
Tang (2017)
Mishra and Rani (2018)
Proposed IVIFCOPRAS
3
Keshavarz Ghorabaee et al. (2014) 4
3
2
4
B2
4
5
3
4
5
2
B3
5
4
5
5
4
5
B4
1
1
1
1
1
1
B5
3
2
2
2
3
3
Spearman’s Correlation rB
0.60
0.40
0.90
0.70
0.30
-
On the basis of comparison with the IVIF-TODIM and other existing approaches, the key outcomes of the developed IVIF-COPRAS approach are as follow: (i)
The IVIF-COPRAS method proposed in this studycan easily handle various practical problems such as plant location problem, composite optimization, AHP-based decision making, radio frequency identification (RFID) network planning, and MAGDM. Whereas the existing methods and IVIF-TODIM cannot effectively perform dissimilarity assessment when the preference points are too large. Therefore, these methods are unable to handle large MCDM problems directly.
(ii)
In the IVIF-COPRAS method, the improved accuracy and score functions are used to tackle the preference attitude of the decision-maker and to make the ranking results more flexible than the other methods.
(iii) In the DANP-based VIKOR method introduced by Kuo et al. (2015), crisp values are used to estimate the HWR facility option. Indeed, the crisp numbers cannot properly model the uncertainty, ambiguity, and incompleteness in uncertain safety and health conditions in the HWR process. In contrast, the proposed IVIF-COPRAS approach is used to rank the HWR option completely inthe IVIF environment, which makes the safety and health assessment in the HWR process more practical and accurate. (iv) The approaches proposed by Hashemi et al. (2015), Kuo et al. (2015), Tang (2017), and the IVIF-TODIM approach have employed only maximizing criteria,and minimizing criteria must be converted into the maximizing criteria before implementation, i.e., the cost-type criteria must be transformed into benefit-type criteria before normalization. However, this procedure is not a minor task. For instance, the transformation in AHP may be the reason for conflicting outcomes, while in the proposed IVIF-COPRAS approach, this deficiency is removed. This is the superiority of IVIF-COPRAS over the others.
Figure 3: Comparison of preference order of HWR facility derived by the proposed method and the existing methods
A comparison between the preference order outcomes derived by the IVIF-COPRAS and the other methods shows a high correspondence isillustrated in Table 12 and Fig. 3. This implies that the ranking deduced by the IVIF-COPRAS should be considered as correct.Thus, the proposed IVIF-COPRAS approach can perform more reliably in handling the correlative MCDM problems. It can also be applied to the MCDM problems in which criteria are independent and have equal importance. 6. Conclusions IVIFSs are appropriate tools to articulate and handle the uncertain and imprecise information in many MCDM problems. The IVIF-information measures have been used in numerous fields such as decision-making, image processing, and medical diagnosis. This paper presented thenovel IVIFdivergence and IVIF-entropy measures based on the exponential function. Comparative examples were presented to illustrate the usefulness of the developed IVIF-information measures over the existing ones. Next, the conventional COPRAS approach was extended to calculate the correlative MCDM problems with IVIFSs. In the present method, the proposed IVIF-information measures are used to compute the criteria weights in the form of Shapley values. Furthermore, some adjustments were
done in the normalization procedure of the conventional COPRAS method. Then, an application related tosafety and health assessment in HWR facility selection wasutilized to illustrate the validity and usefulness of the proposed approach with IVIFSs. To validate the outcomes of the proposed method, a comparison was made between the developed IVIF-COPRAS and different existing approaches. The results confirmed that the developed approach was well-organized, easily applicable, andefficient. The advantages of the IVIF-COPRAS approach are the computation simplicity in IVIFSs, implementation of a process for calculating more realistic criteria weights, and increasing the stability of the approach. In the future, the developed MCDM model can be further extended to PFSs, interval-valued Pythagorean fuzzy sets, HFSs and interval-valued hesitant fuzzy sets. Moreover, the authors can expand our research using different MCDM platforms (for instance, AHP, Combined Compromise Solution (CoCoSo), ELECTRE, and MABAC) in order to select the most appropriate HWR facility selection. Appendix A A.1. Proof of Theorem 4.1 (D1). It is evident from Definition 3.5 that I S , T I T , S . (D2). It can easily be verified that I S, T 0 iff S T . (D3). From (13), it is also evident that I S , T I S c , T c and I S c , T I S , T c . (D4). As 0 mS (ui ), nS (ui ) 1, 0 ( 1), then, we have I S , T 0 , and I S , T 0 iff S T . The
I S , T attains
measure
the
maximum
value
when
m (u ), m (u ) , n (u ), n (u ) 1,1 , 0, 0 and m (u ), m (u ) , n (u ), n (u ) 0,0 ,1,1 , S
i
S
i
S
i
S
i
T
i
T
i
T
i
T
i
which follows that I S , T p ln 4. When T S c , then I S , T attains its maximum value and when S T , then I S , T has minimum value, therefore, the value of proposed divergence measure lies between p ln 4. For the process of normalization, we divided the I S , T by p ln 4 , we obtain 0 I S, T 1.
(D5). Consider S T W , then m S (u i ) mT (u i ) mW (u i ), m S (u i ) mT (u i ) mW (u i ),
nS (ui ) nT (ui ) nW (ui ) and nS (u i ) nT (u i ) nW (u i ). Now,
m (u ) 1 nS (ui ) mT (ui ) 1 nT (ui ) 0 S i 2 2 1 i
m (u ) 1 nS (ui ) mW (ui ) 1 nW (ui ) S i 1, 1( 0). (A1) 2 2 2 i
Let i exp i , then 1i 2i .
(A2)
This implies I S , T 1i 2i I S , W . i
i
Similarly, I T , W I S , W . (D6). To prove (D6), we partition U into eight subsets as follows: U ui U | S ui T ui W ui ui U | S ui W ui T ui ui U | S ui T ui W ui ui U | S ui W ui T ui ui U | T ui R ui W ui ui U | T ui W ui S ui ui U |W ui S ui T ui ui U |W ui T ui S ui , (A3)
which are denoted by 1 , 2 ,..., 8 . From Montes et al. (2002), for each j ; j 11 8,
and
S W ui T W ui S ui T ui
(A4)
S W ui T W ui S ui T ui .
(A5)
Therefore, from (D5), we obtain
I ( S W , T W ) I ( S , T ) for every W IVIFSs (U ). (D7). The proof is similar to (D6). A.2. Proof of Theorem 4.2
To show (24) a valid measure of IVIF entropy, it must hold the postulates (EN1)-(EN4) of Definition 3.4. The measure (18) can be written as
H1 S H S , (A6) where
H S
p 1 m ( u ) ln i S p ln 4 i 1
nS (ui )ln
nS ( ui ) mS ( ui ) nS ( ui )
m S ( u i ) m S
( ui )
n S
( ui )
n (u )ln S
i
m
S
( u i ) ln
nS ( ui ) mS ( ui ) nS ( ui )
m S ( u i ) m S
( u i ) n S ( u i )
(u )ln 2 (u )ln 2 S
i
S
i
(A7)
is an entropy measure introduced by Chen et al. (2010) and function ui : 0,1 [0,1] is defined as
ui p 41 1 exp 1 ui 1 .
(A8)
Here, function ui is increasing at ui 0. Therefore, H1 S is an increase of H S and H S is the IVIF-entropy. Hence, it is lucid that H1 S is the valid parametricIVIF-entropy. A.3. Proof of Theorem 4.3 To prove (18) asa valid measure of IVIF-entropy, it must hold the postulates (EN1)-(EN4) given in Definition 3.4. (EN1). Suppose that S be a crisp set, i. e., mS (ui ), mS (ui ) 1,1 , nS (ui ), nS (ui ) 0, 0 or
mS (ui ), mS (ui ) 0, 0 , nS (ui ), nS (ui ) 1,1 , for each ui U . Therefore, H 2 S 0. If H 2 S 0, then each term of (18) is non-negative. It implies that it should be equal to zero, i.
e., min mS (ui ), nS (ui ) 0, min mS (ui ), nS (ui ) 0, S (ui ) 0 and S (ui ) 0, for each ui U . Thus, S is a crisp set. (EN2). Let mS (ui ), mS (ui ) nS (ui ), nS (ui ) , i. e., mS (ui ) nS (ui ), mS (ui ) nS (ui ), for all ui U . From (18), we get H2 (S ) 1. Now, assume that H 2 ( S ) 1, then for all ui U , we have
1 max m ( u ), n ( u ) 1 S i S i 2 min ( ), ( ) m u n u max m ( u ), n ( u ) S i S i 2 S i S i p S ( ui ) S ( ui ) 1 min mS (ui ), nS (ui ) e p 1 i 1 S (ui ) S (ui )
1 min m ( u ), n ( u ) 12 max mS (ui ), nS (ui ) 2min mS (ui ), nS (ui ) S i S i S ( ui ) S ( ui ) max mS (ui ), nS (ui ) e 1. (u ) (u ) S i S i
1 12 min mS (ui ), nS (ui ) 2 max mS(ui ), nS(ui ) max mS ( ui ), nS ( ui ) S ( ui ) S ( ui ) It implies that min mS (ui ), nS (ui ) e (u ) (u ) S i S i
1 min m ( u ), n ( u ) 12 max mS (ui ), nS (ui ) 2min mS (ui ), nS (ui ) S i S i S ( ui ) S ( ui ) max mS (ui ), nS (ui ) e 0. (u ) (u ) S i S i
Thus, we have concluded that mS (ui ), mS (ui ) nS (ui ), nS (ui ) , for all ui U . (EN3). It is clear that
S c ui , mS (ui ), mS (ui ) , nS (ui ), nS (ui ) : ui U . Therefore, from (18), we get
H 2 ( S ) H 2 ( S c ). (EN4). Suppose that mT (ui ) nT (ui ), mT (ui ) nT (ui ), for each ui U and S T , i. e.,
mS (ui ) mT (ui ), mS (ui ) mT (ui ), nS (ui ) nT (ui ), nS (ui ) nT (ui ), for each ui U . Then, it follows that mS (ui ) nS (ui ) and mS (ui ) nS (ui ). From (18), we have 12 mS (ui ) mS (ui ) 1 H 2 (S ) p i 1 S (ui ) S (ui ) p
1 n ( u ) n ( u ) 2 S i(u ) S i (u ) e S i S i
12 nS (ui ) nS (ui ) (u ) (u ) S i S i
1 m ( u ) m ( u ) 2 S (i u ) S (iu ) e S i S i .
Therefore, p
H 2 ( S ) 1p 12 (2 nS (ui ) nS (ui )) e
1 (2 m ( u ) m ( u )) S i S i 2
i 1
1 (2 n (u ) n (u )) 12 (2 mS (ui ) mS (ui )) e 2 S i S i and H 2 (T )
p
1 p
i 1
1 2
( 2 n T ( u i ) n T ( u i )) e
12 ( 2 m T ( u i ) m T ( u i )) e
1 2
1 2
( 2 m T ( u i ) m T ( u i ))
( 2 n T ( u i ) n T ( u i ))
.
Since S T , we have nS (ui ) nS (ui ) nT (ui ) nT (ui ) and mS (ui ) mS (ui ) mT (ui ) mT (ui ). Therefore, H 2 ( S ) H 2 (T ). Similarly, if mT (ui ) nT (ui ), mT (ui ) nT (ui ), for each ui U and S T , then we can easily prove that H 2 ( S ) H 2 (T ).
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Highlights
1. New divergence and entropy measures are introduced for IVIFSs and compared with existing measures. 2. An interval-valued intuitionistic fuzzy COPRAS is developed. 3. The entropy and Shapley function based procedure for criteria weights is constructed. 4. The proposed approach is applied for an HWR facility selection problem. 5. Comparative work and sensitivity analysis from the case study are given.