In order to make comparisons between two preference information that are expressed in IFS, some metric methods were introduced, that is the score functions and accuracy functions (Chen & Tan, 1994; Hong & Choi, 2000; Wu, 2015; Xu, 2007) which were applied for solving MADM problems. However, investigations of these functions show some vital shortcomings, according to Wu (2015) the results obtained using the score functions and accuracy functions are not always consistent, also they often produce negative priority vector in their applications. Although, the exponential score function later proposed by Wu, (2015) as a remedy, appears to address these shortcomings, the function is only effective for determining priority weight that involves pairwise comparison. Therefore, in this paper, we propose a new exponential related function (ER) and develop an intuitionistic fuzzy TOPSISmodel based on the exponential-related function (IF-TOPSIS EF ) to solve MADM problems in which the
D ECISION-MAKING means selecting the best alterna- tives from the feasible alternatives. With the develop- ment of science, decision-making extends from the single attribute to multiple attributes. In order to make a proper decision-making and apply decision-making in the actual situation, Wang et al. proposed a single-valued neutro- sophic set (SVNS) and also introduced the set-theoretic operators of SVNSs. Sodenkamp et al. developed a novel approach that utilizes single-valued neutrosophic sets (NSs) to process independent multi-source uncertainty measures affecting the reliability of experts assessments in group multi- criteria decision-making (GMCDM) problems. Abdelbasset et al. modeled the imperfections of various data in smart cities and then proposed a general framework for processing imperfect and incomplete information through using SVNS and rough set theories. On the other hand, many multi- attributedecision-making(MADM) methods based on SVNS have been put forward for solving MADM problems.
It seems that the classical TOPSIS is a good method for solving the MADM problem, but there are many emerging methods to be put forward in recent years. Pamuca and Cirovic initiated a Multi-Attributive Border Approximation area Comparison (MABAC) method to solve MCDM problem. Peng and Yang ,  proposed the Pythagorean fuzzy Choquet integral average operator for solving MAGDM. Xue et al.  proposed a MABAC method for material selection and used the extended group decision method to sort the alternative materials. Sun et al.  developed a project-based MABAC method with fuzzy language item sets and extended this method into hesitant fuzzy linguistic environment.
TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) is proposed by Hwang and Yoon , and it is a popular approach to MCDM problems. The basic principle is that the chosen alternative should have the shortest distance from the positive ideal solution and the farthest distance from the negative ideal solution. In the TOPSIS, the performance ratings and the weights of the criteria are given as crisp values. In many cases, crisp data are inadequate to model real life situations. Jahanshahloo et al  extends the TOPSIS method to the fuzzy decisionmaking situations by considering interval numbers and defining crisp Euclidean distance between two interval numbers. Wang and Elhag proposes a fuzzy TOPSIS method based on alpha level sets and presents a nonlinear programming
Literature - has conducted the research to the fuzzy language multi-objective decisionmaking question: In literature , the linguistic variable of the evaluation value and weight vectors is modeled by the normal fuzzy number, and the decisionmaking framework based on the linguistic operator is established by the weighted mean method. The linguistic operator based on normal fuzzy numbers is given, and the calculating method for the cuts of the comprehensive evaluation value is put forward. The fuzzy number of evaluation is expressed by discrete cuts, and the normal fuzzy number is aggregated. In literature , defines the following concept: the expectation level of alternative, the uncertain linguistic negative point, the achievement scale, the alternative comprehensive scale under uncertain linguistic environment. Based on these concepts, some linear programming models are established, through which the decision maker interacts with the analyst. In literature , A TOPSIS method is proposed to deal with multiple attributedecision-making [MADM] problems with attribute weights unknown completely and the attribute values taken the form of uncertain linguistic variables. In literature , with respect to multiple-attributedecision- making (MADM) problems with uncertain linguistic information, a decision analysis method based on linguistic probability is proposed. The concept to uncertain linguistic variable is introduced and linguistic probabilistic ordered weighted averaging (LPOWA) operator is proposed. In literature , study the multiple attributedecisionmakingproblems, in which the
In conclusion, DEA ranking models (such as AP, LJK and models with restriction weights) and Cook-Kress model (which uses both quantitative and qualitative criteria) have more discrimination power than MADM models (SAW and TOPSIS) especially in big problems. MADM models are so close compare to DEA models and eventually valid ranking of alternatives is questionable. Another important point should be expressed that there is no generally accepted method to make a relative comparison of DEA models or any other MADM models among themselves. However, comparing of ranking results has been performed by determining of Kendal correlation coefficient and finally results of DEA models and Cook- Kress model have good correlation with MADM models. So these models can be used to solve MADM problems. In addition to this, models with restriction weights because of applying value judgments are the best choice for solving MADM problems. In other words, incorporating decision maker or manager preferences enhances the correlation between DEA and MADM models.
In this section, in the first attempt, the Dantzig-wolf decomposition method is successfully applied to decompose the original problem into the q independent linear sub problems. In other words, the L-dimensional problem space is reduced to a one-dimensional space by applying the Dantzig-Wolfe decomposition algorithm. Appling new compromised method; the objectives of each sub problem are aggregated. To this mean, the individual positive ideal solution (PIS) and negative ideal solution (NIS) are calculated for each objective. Applying PIS and NIS, the bi- objective problems are constructed for jth sub problem. Finally, the final single objective problem is solved to obtain final optimal solution. The proposed method has the following steps: Step 1. Applying the Dantzig-wolf ecomposition method, decompose the primal problem into q independent sub problems for all objective functions and constraints to reduce the dimension of primal problem.
For the common multi-attributedecisionmaking (MADM) problems under the hesitant fuzzy en- vironment, there are many mature solutions, such as hesitant fuzzy TOPSIS method  and hesitant fuzzy TODIM method  which are often used to deal with the energy policy selection problem  and service quality evaluation . But those methods based on static information at a certain time will not be applicable when the actual situation is constantly changing. For example, in the early stage of outbreak of the Novel Coronavirus (COVID-19) in 2019 1 , the virus was not particu- larly harmful and infectious, but with the virus spread rapidly, the corresponding problems related to politics, economy, culture and other aspects had also been changing rapidly. In this situation, the static decision-making methods cannot grasp the dynamic change of the events and the corresponding changes in the attribute values cannot be well reflected, by contrast, the dynamic MCDM methods is more appropriate.
Liang and Wang (1993) suggested a fuzzy multi-criteria decision-making approach for industrial robot selection. They suggested group decisionmaking for the selection of robots. Zhao et al. (1996) introduced genetic algorithm (GA) for optimal Robot Selection problem in a CIM system. Goh (1997) used AHP method for robot selection incorporating inputs from multiple decision makers and considered both the subjective and the objective criteria. Chu and Lin (2003) used a fuzzy TOPSIS Method for robot selection with subjective as well as objective criteria. Bhangale et al. (2004) used TOPSIS and graphical method for the selection of a robot for some pick-n-place operation. Shih (2008) suggested an incremental analysis method with group TOPSIS for the selection of industrial robots. Chatterjee et al. (2010) applied ‘VIsekriterijumsko KOmpromisno Rangiranje’ (VIKOR) and ‘ELimination and Et Choice Translating REality’ (ELECTRE) methods for the selection of robot for some industrial application.
A good amount of research work has been carried out in recent past for robot selection, but still it is a new concept in industry at large, so it is not unusual for an industry to be a first time robot purchaser. Knott and Getto (1982) suggested a model to evaluate different robotic systems under uncertainty, and different alternatives were evaluated by computing the total net present values of cash flows of investment, labor components, and overheads. Liang and Wang (1993) proposed a robot selection algorithm by combing the concepts of fuzzy set theory and hierarchical structure analysis. The algorithm was used to aggregate decision makers‟ fuzzy assessments about robot selection attribute .Agrawal et al. (1991) proposed a robot selection procedure to rank the alternatives in a shortlist by employing TOPSIS (technique for order preference by similarity to ideal solution) method. An expert system was also developed to assist the decision maker to establish priorities and visualize the selection process at various stages. Zhao et al. (1996) have introduced genetic algorithm (GA) for optimal Robot Selection problem for a CIM system. Khouja and Kumar (1999) used options theory and an investment evaluation procedure for selection of robots weightings, and to obtain fuzzy suitability indices. Bhangale et al. (2004) had used TOPSIS and graphical method for the selection of a robot for some pick-n-place operation.Bhattacharya et al. (2005) integrated AHP and quality function deployment (QFD) methods for solving industrial robot selection problems, while considering seven technical requirements and four alternative robots.
In the fuzzy set theory  there were no scopes to think about the hesitation in the membership degree, which arise in various real life situations. To overcome these situations Atanassov  introduced theory of intuitionistic fuzzy set in 1986 as a generalization of fuzzy set.Most of the problems in engineering, medical science, economics, environments etc have various uncertainties. Molodtsov initiated the concept of soft set theory as a mathematical tool for dealing with uncertainties. Research works on soft set theory are progressing rapidly. Maji et al. defined several operations on soft set theory. Combining soft sets with fuzzy sets and intuitionistic fuzzy sets, Feng et al. and Maji et al.[9,10] defined fuzzy soft sets and intuitionistic fuzzy soft sets which are rich potentials for solvingdecisionmaking problems.Matrices play an important role in the broad area of science and engineering. The classical matrix theory cannot solve the problems involving various types of uncertainties. In  Yang et al, initiated a matrix representation of a fuzzy soft set and applied it in certain decisionmakingproblems. The concept of fuzzy soft matrix theory was studied by Borah et al. in . In , Chetia et al. and in  Rajarajeswari et al. defined intuitionistic fuzzy soft matrix.Again it is well known that the matrices are important tools to model/study different mathematical problems specially in linear algebra. Due to huge applications of imprecise data in the above mentioned areas, hence are motivated to study the different matrices containing these data. Soft set is also one of the interesting and popular subject, where different types of decisionmaking problem can be solved. So attempt has been made to study the decisionmaking problem by using intuitionistic fuzzy soft aggregation operator. Das and Kar  proposed an algorithmic approach for group decisionmaking based on IF soft set. The authors  have used cardinality of IF soft set as a novel concept for assigning confident weight to the set of experts. Cagman and Enginogh[3 ,4] pioneered the concept of soft matrix to represent a soft set. Mao et al. presented the concept of intuitionistic fuzzy soft matrix(IFSM) and applied it in group decisionmaking problem.
The linear programming technique for multidimensional analysis of preference (LINMAP) is one of the well-known methods for multiple attribute group decisionmaking (MAGDM). In this paper, we extend the LINMAP method to solve MAGDM problems in interval-valued intuitionistic fuzzy environment in which all the preference information provided by the decision- makers is presented as interval-valued intuitionistic fuzzy decision matrices and the preference information about the alterna- tives is completely unknown. We introduce two deﬁnitions of distances between interval-valued intuitionistic fuzzy sets. To calculate the weights of attributes we develop a new linear programming model based on the group consistency and incon- sistency index deﬁned on the basis of pairwise comparison preference relations on alternatives given by the decision makers, the weighted distance of each alternative to the interval-valued intuitionistic fuzzy positive ideal solution can be calculated to determine the ranking order of all alternatives for the decision makers, and the ranking order of alternatives and the best alternative(s) for the group are generated by using the Borda’s function.
Due to the essence of optimizing multiple objectives in PFSSP, it is also extended to the multi-objective domain with many challenging approaches (non-heuristic and meta-heuristic). Selen and Hotts  solved a multi- objective flowshop scheduling problem (MOPFSSP) with m-machines by formulating a mixed-integer goal programming model with two objectives that is makespan and mean flowtime. Wilson  proposed an alternative model for it, by considering a fewer number of variables but at the same time he has added large number of constraints to it. Both the models have included same number of integer variables. Daniels and Chambers  proposed a branch and bound approach with two objectives (makespan and maximum tardiness) where they computed the Pareto solution for a 2-machine flowshop scheduling problem. Rajendran  also presented a similar procedure along with two heuristic approaches for the 2-machine flowshop scheduling problem with two objectives: minimization of TFT subject to optimal makespan. Similarly two different methodologies (one is based on a Branch and Bound (B&B) technique of exact algorithms and other one is based on Palmer approach of heuristic algorithms) are used  to find the optimum solution for minimization of bi-criterion (makespan, weighted mean flowtime) objective function of three machines FSSP with transportation times and weight of the jobs. Recently a production scheduling problem in hybrid shops has been solved by Mousavi et al., by assuming some realistic assumptions.
Multiple-AttributeDecisionMaking (MADM) methods have been extensively applied to various areas, such as society, management science, economics, military research and public administration [ 1–5 ]. However, most MADM methods focus on decisionmakingproblems at the same period, such as those proposed by Ye [ 6 ] who developed a MADM model with interval-valued, intuitionistic, fuzzy numbers, and Jaskowski et al. [ 7 ] who presented an extended fuzzy AHP model for group decisionmaking, at the same period. Greco et al. [ 8 , 9 ] and Blaszczynski et al. [ 10 ] extended the rough set theory into a multi-attributedecisionmaking method, and Hu et al. [ 11 ] also extended a rough set MADM model to solve a multi-attributedecisionmaking problem at the same period.
some decisionproblems. Then some aggregation operators based on these were proposed by Xu [5-6] and some methods for MADM with IFS were proposed in [7-8]. Furthermore, Atanassov and Gargov [4,9] extended the membership function and non-membership function to interval numbers and proposed interval-value intuitionistic fuzzy set (IVIFS). But IFS and IVIFS can only deal with incomplete information, but not uncertain and inconsistent information.
Fuzzy model is also used to select a project for research and development (R & D) with multi-criteria decisionmaking. The project selection used several qualitative and quantitative criteria. The criteria include cost and some of the obtained advantages if the project was implemented. However, models produced by Ramadan  still can not be used in group.In order to anticipate a group assessment, Zhou et al.  implemented fuzzy logic in decision support system to assess project produced by students. The project is rated by more than one person with several fuzzy criteria. The best project is a project with the highest membership value. Another method for the decision support system is analytical hierarchy process (AHP) fuzzy. AHP fuzzy can help users to make decisions on both structured and semi structured problems . In addition,  fuzzy analytical hierarchy process is also used to help make decisions on the process of multicriteria robot selection. Researchers  have described several procedures on a modified technique for order preference by similarity to ideal solution (TOPSIS) method so that the TOPSIS can also be used for a case of decision made in group or multi-criteria group decisionmaking (MCGDM). In this study, TOPSIS algorithm is used in FMADM to asses the eligibility of scholarship recipients and helping the decision maker to make a quick, accurate and objective decision.
However, apart from that Technique for Or- der Preference with respect to the Similarity to the Ideal Solution (TOPSIS), developed by Hwang and Yoon , there is a well-known Multi-Criteria Decision-Making (MCDM) method. The aim of this method is to choose the best alternative whose distance from its positive ideal solution is the shortest. After their existence, numerous attempts are made by the researchers to apply the TOPSIS method under the fuzzy and IFS environment. For instance, Szmidt and Kacprzyk  dened the concept of distance measure between the IFSs. Hung and Yang  presented the similarity measures between the two dierent IFSs based on Hausdor distance. Boran et al.  applied the TOPSIS method to solve the problem of human resource personnel selection. Dugenci  presented a distance measure for IVIF set and their application to MCDM with incomplete weight information. Garg  presented a generalized improved score function for IVIFSs and their TOPSIS-based method for solving the DM problems. Mohammadi et al.  presented a gray relational analysis and TOPSIS approach to solving the DM problems. Garg et al.  presented a generalized entropy measure of order and degree under the IFS environment and applied it to solve the DM problems. Biswas and Kumar  presented an integrated TOPSIS approach for solving the DM prob- lems with IVIFS environment. Vommi  presented a TOPSIS method using statistical distances to solve DM problems. Singh and Garg  developed the distance measures between the type-2 IFS. Li  presented a nonlinear programming methodology-based TOPSIS method for solvingMulti-AttributeDecisionMaking (MADM) problems under IVIFS environment. Garg and Arora  extended the Li  approach to the interval-valued intuitionistic fuzzy soft set environ- ment. Lu and Ye  developed logarithm similarity measures to solve the problems under interval-valued fuzzy set environment. Garg and Kumar  presented new similarity measures for IFSs based on the connec- tion number of the set pair analysis. Askarifar et al.  presented an approach to studying the framework of Iran's seashores using TOPSIS and best-worst MCDM methods. In [33,34], the authors developed a group DM method under IVIF environment by integrating extendedTOPSIS and linear programming methods. Kumar and Garg [35,36] presented the TOPSIS ap- proach for solving DM problems by using connection number of the set pair analysis theory.
Hesitant fuzzy set (HFS), which allows the membership degree of an element to be a set of several possible values, it has attracted more and more attention due to its powerfulness in representing uncertainty. In this paper, we proposed an approach based on TOPSIS and entropy-weighted method for solvingmulti-attributedecisionmaking (MADM) problems under hesitant fuzzy environment and the attribute weights is complete unknown. First, we introduce the basic concepts of HFSs. Then, we determine the attribute weights through entropy- weighted method under hesitant fuzzy information. Then, the similarity degree of every alternative with hesitant fuzzy positive ideal solution is displayed to rank all the alternatives. Finally, a numerical example is given to illustrate the effectiveness and feasibility of the proposed method.
In this paper, we explain the value of both Trapezoidal and Triangular Fuzzy Numbers and develop a new ranking method based on the value of fuzzy number which in turn will be very helpful in decisionmaking situations. Then we propose a Multi Criteria DecisionMaking (MCDM) Model based on the proposed ranking method. Arithmetic mean operation of fuzzy numbers is used for aggregating experts’ judgments.
Two-level multiple objectives decisionmakingproblems consist of the objectives of the leader at its first level and that is of the follower at the second level. The decision maker (DM) at each level attempts to optimize his individual objectives, which usually depend in part on the variables controlled by the DM at the other levels and their final decisions are executed sequentially where the first level decision maker (FLDM) makes his decision firstly. The research and applications concentrated mainly on two-level programming (see f. i. [ 7, 8, 11, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 34,35,36, 51,53,56,57]).