Robot Evaluation and Selection Using the Hesitant ...

Journal of Testing and Evaluation

doi:10.1520/JTE20170094

available online at www.astm.org

Hu-Chen Liu,1,2 Hao Zhao,2 Xiao-Yue You,2,3 and Wen-Yong Zhou4

Robot Evaluation and Selection Using the Hesitant Fuzzy Linguistic MULTIMOORA Method Reference Liu, H.-C., Zhao, H., You, X.-Y., and Zhou, W.-Y., “Robot Evaluation and Selection Using the Hesitant Fuzzy Linguistic MULTIMOORA Method,” Journal of Testing and Evaluation https://doi.org/10.1520/JTE20170094. ISSN 0090-3973

ABSTRACT Manuscript received February 13, 2017; accepted for publication September 26, 2017; published online xxxx xx, xxxx. 1

School of Management, Shanghai University, 99 Shangda Rd., Shanghai 200444, PR China, https://orcid.org/ 0000-0003-4566-2107

With the development of modern technology, industrial robots have been applied extensively in different industries to perform high-risk jobs and produce highquality products. However, selecting an appropriate robot for a specific manufacturing environment is a difficult task for decision makers because of the increase in complexity, production demands, and the availability of different robot types. Normally, robot selection can be regarded as a complex multicriteria decision-making problem, and decision makers often use uncertain linguistic terms to express their assessments because of time pressure, lack of data, and their

2

School of Economics and Management, Tongji University, 1239 Siping Rd., Shanghai 200092, PR China, https://orcid.org/ 0000-0001-7142-9478 (H.Z.)

limited expertise. In this article, a modified MULTIMOORA (Multiobjective Optimization by Ratio Analysis plus the Full Multiplicative Form) method based on hesitant fuzzy linguistic term sets (named HFL-MULTIMOORA) is proposed for evaluating and selecting the optimal robot for a given industrial application. This

3

4

Institute for Manufacturing, University of Cambridge, 17 Charles Babbage Rd., CB3 0FS Cambridge, United Kingdom School of Economics and Management, Tongji University, 1239 Siping Rd., Shanghai 200092, PR China (Corresponding author), e-mail: [email protected]

method deals with the decision makers’ uncertain assessments with hesitant fuzzy linguistic variables, which can increase the flexibility of representing linguistic information. Finally, an empirical example is presented to demonstrate the proposed method, and the results indicate that the HFL-MULTIMOORA provides a useful and practical tool for solving robot selection problems within a hesitant linguistic information environment. Keywords robot systems, multiobjective optimization by ratio analysis method, hesitant fuzzy linguistic term sets, robot selection

Copyright © 2018 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959

LIU ET AL. ON ROBOT SELECTION USING HFL-MULTIMOORA

Introduction “Made in China 2025” is the first ten-year action plan designed to transform China from a manufacturing giant into a world manufacturing power [1,2]. Meanwhile, Europe is talking about the fourth industrial revolution, which requires internet of things as the basic information platform [3,4]. Robots can serve as a significant intellectual support to these advanced manufacturing models. They are recognized by manufacturing firms as an important technological advance because the automatic workers can help factory owners improve product quality and increase productivity. Industrial robots are considered to be automatically controlled, general-purpose, and reprogrammable manipulators to conduct different tasks [5]. The recent growth of microelectronics, information, and automation technologies is the key reason for the increased utilization of robots with distinct capabilities, features, and specifications in different advanced manufacturing systems. Robots are able to address some safety and health issues in the workplace and can also be used to perform high-risk jobs. For example, they can help people conduct research in unexplored areas, defuse bombs, and clean the exterior of skyscrapers. Therefore, manufacturers prefer to use robots in the practical industrial applications where repetitive, difficult, or hazardous tasks need to be performed [6,7]. But when decision makers, particularly the first-time robot purchasers, implement an improper selection of robots, the company’s competitiveness will be adversely affected [8]. Consequently, selecting the right robot from a wide variety of feasible alternatives in the real-time manufacturing market has always been one of the most challenging tasks for manufacturing companies [9–11]. In recent years, various approaches have been developed by researchers for robot evaluation and selection. However, most of the current robot selection models have some serious limitations because of the fact that they assess the performance of alternative robots by using single linguistic terms rather than following the information provided by decision makers. Rodríguez, Martínez, and Herrera [12] proposed the concept of hesitant fuzzy linguistic term sets (HFLTSs), which provides a new and more powerful way to represent decision makers’ assessments in the decision-making process. The HFLTSs can be applied to multicriteria linguistic decision-making problems in which decision makers give their judgments by using linguistic expressions based on either comparative terms, such as “between very low and medium”, or simple terms, such as “very low, medium, and very high”. The merit of the HFLTS approach is that it can increase the flexibility and capability of eliciting and representing experts’ linguistic assessment information. Because of time pressure and lack of knowledge and data, decision makers are often hesitant in their preferences during the robot selection process. Hence, the hesitant fuzzy linguistic representation approach offers a more flexible and precise way for managing linguistic assessments in the robot evaluation and selection. To select the most suitable robot, it is necessary to balance many objective and subjective factors, some of which are conflicting in nature and have different units. In addition, there is a need for systematic and efficient methods to assist decision makers in evaluating and selecting robots because the selection of an optimal robot for a specific application is a complicated multicriteria decision-making (MCDM) problem [6,7,13]. The MULTIMOORA (Multiobjective Optimization by Ratio Analysis plus the Full Multiplicative Form) method, proposed by Brauers and Zavadskas [14], is an effective and comprehensive method to deal with MCDM problems. It consists of three parts including the ratio system, the reference point, and the full multiplicative form. The MULTIMOORA and its extended forms have been widely applied in numerous fields



for solving real-life decision-making problems, such as materials selection [15,16], failure mode and effect analysis [17,18], personnel selection [19,20], and public investment analysis [21]. However, the use of the MULTIMOORA within the robot selection framework has not been accomplished before. Therefore, this article aims to develop an extended MULTIMOORA method with HFLTSs to solve the robot selection problem under an uncertain linguistic information context. The rest of this article is organized as follows: the section titled “Literature Review” provides an overview of previous methods for robot selection and applications of the MULTIMOORA. The section titled “Preliminaries” briefly presents some basic concepts and operational laws of HFLTSs. In the section immediately following “Preliminaries,” an extended MULTIMOORA approach is developed to solve the robot selection problem with hesitant fuzzy linguistic information. In “An Illustrative Example,” a practical example is given to illustrate the applicability and advantages of our proposed model. Finally, the last section summarizes the work and draws some suggestions for further research.

Literature Review METHODS FOR ROBOT SELECTION

During the last several decades, a variety of methodologies have been developed to solve robot selection problems. For instance, Kumar and Garg [8] presented a deterministic quantitative model using the distance-based approach to evaluate and select the best robot. Chatterjee, Athawale, and Chakraborty [22] used the VIKOR (Visekriterijumsko Kompromisno Rangiranje) and ELECTRE (Elimination and Et Choice Translating Reality) methods for robot selection and compared their relative performance for a given industrial application. Kentli and Kar [23] proposed a decision model for industrial robot selection based on the concepts of satisfaction function and distance measure. Sen et al. [24] applied the preference ranking organization method for enrichment evaluation (PROMETHEE) II method in relation to the robot selection problem, subjected to a set of quantitative (objective) evaluation data. In many practical robot selection processes, decision makers often use linguistic terms to express their subjective assessments and there may exist uncertain and hesitant linguistic information. Therefore, to deal with the vagueness and ambiguity in robot selection, Rao, Patel, and Parnichkun [5] presented a subjective and objective integrated MCDM method and used fuzzy logic to convert qualitative attributes into quantitative attributes. A fuzzy digraph method is proposed by Koulouriotis and Ketipi [25] for the evaluation and selection of robots, where the subjective criteria are assessed in fuzzy linguistic terms. Tansel İç, Yurdakul, and Dengiz [26] developed a two-phase robot selection model, namely ROBSEL, in which the user can obtain a suitable set of robots according to the need of an application and rank the robots by using fuzzy analytical hierarchy procedure (AHP) method. In addition, Chu and Lin [27] proposed a fuzzy technique for order preference by similarity to ideal solution (TOPSIS) method, where the ratings of alternatives against subjective criteria are expressed in linguistic terms, represented by fuzzy numbers. Vahdani et al. [28] proposed an interval-valued fuzzy TOPSIS-based robot selection method that can reflect both subjective judgments and objective information in realistic circumstances. Liu et al. [13] presented an interval 2-tuple linguistic TOPSIS method considering both subjective judgments and objective information to handle robot selection problems in uncertain and incomplete information environments.



Recently, Ghorabaee [7] proposed an approach based on the VIKOR method with interval type-2 fuzzy numbers to handle the robot selection problem. Parameshwaran, Praveen Kumar, and Saravanakumar [11] presented an integrated approach using fuzzy AHP, fuzzy TOPSIS/fuzzy VIKOR, and Brown-Gibson models for robot selection and provided an illustration with a case study related to the selection of a robot for teacher purposes. Gitinavard, Mousavi, and Vahdani [29] researched a soft computing approach based on the interval-valued hesitant fuzzy complex proportional assessment (IVHF-COPRAS) method that could be used toward the selection of industrial robots under uncertainty. Sen, Datta, and Mahapatra [10] extended the classical PROMETHEE method in a fuzzy environment so as to solve the robot selection decision-making problems that involve both objective as well as subjective data. An integrated linguistic MCDM model was developed by Xue et al. [6] for robot evaluation and selection with incomplete weight information, in which hesitant 2-tuple linguistic term sets were used to capture imprecise assessment information of decision makers and an extended qualitative flexible multiple criteria approach was utilized to select the most suitable robot for a particular industrial application. APPLICATIONS OF MULTIMOORA

Since its introduction, the standard MULTIMOORA has been extended by many scholars to model and manage MCDM problems within different decision-making environments. For instance, Gou et al. [30] developed a double hierarchy hesitant fuzzy linguistic MULTIMOORA method to evaluate the implementation status of haze-controlling measures in China. Zhao, You, and Liu [31] presented an approach based on continuous entropy weights and the MULTIMOORA method for failure mode and effect analysis under the interval-valued intuitionistic fuzzy context. Tian et al. [32] proposed an improved MULTIMOORA approach that takes advantage of linguistic scale functions, aggregation operators, and distance-based methods for dealing with neutrosophic linguistic MCDM problems. Hafezalkotob, Hafezalkotob, and Sayadi [15] suggested an extension of the MULTIMOORA method using interval numbers for materials selection, and Hafezalkotob and Hafezalkotob [33] extended the MULTIMOORA approach based on the Shannon entropy concept under a fuzzy environment to solve material selection problems. Çebi and Otay [34] proposed a two-stage fuzzy approach for addressing supplier selection and order allocation problems, in which fuzzy MULTIMOORA was used to evaluate and select suppliers and fuzzy goal programming was adopted to determine the amount of order allocated to the selected suppliers. In Ref. [35], the authors evaluated healthcare waste treatment technologies with a hybrid MCDM model integrating 2-tuple decision making trial and evaluation laboratory and fuzzy MULTIMOORA methods. In Ref. [36], a supplier selection method was provided by extending the MULTIMOORA with the 2-tuple linguistic representation method. Additionally, other meaningful extensions of the MULTIMOORA technique in previous research include the neutrosophic MULTIMOORA [37], the trapezoidal intuitionistic fuzzy MULTIMOORA [19], and the interval 2-tuple linguistic MULTIMOORA [38]. The previous literature review shows that, although a number of extensions of the classical MULTIMOORA have been developed, very little research has been performed to extend the MULTIMOORA method within uncertain linguistic contexts (e.g., Liu et al. [38]). However, our proposed modified MULTIMOORA method based on hesitant fuzzy linguistic term sets (HFL-MULTIMOORA) has its advantages that differ from the current methods. Firstly, alternative robots are evaluated using HFLTSs, which can handle both the hesitance and divergence of decision makers’ subjective assessments. Secondly, we use the concept of statistical variance to determine the importance coefficients of evaluation



criteria; this is a remarkably excellent method for acquiring objective criteria importance coefficients. Finally, we apply the proposed method to robot selection and use a case study in an auto company to demonstrate its effectiveness and practicality.

Preliminaries HFLTSs

The theory of HFLTSs was introduced first by Rodríguez, Martínez, and Herrera [12] based on the fuzzy linguistic approach [39] and hesitant fuzzy sets [40,41] to represent comprehensive linguistic expressions. Next, the basic concepts and operations of HFLTSs from Rodríguez, Martínez, and Herrera [12] are reviewed. Definition 1 Let S = fs0 , s1 , : : : , sg g be a linguistic term set. An HFLTS H s is an ordered finite subset of consecutive linguistic terms of the linguistic term set S. The HFLTS can be used to elicit several linguistic values for a linguistic variable, but it is not similar to a human way of thinking and reasoning. To make it more applicable, Rodríguez, Martínez, and Herrera [12] proposed a context-free grammar GH to generate simple but elaborated linguistic expressions. Definition 2 Let GH = ðV N , V T , I, PÞ be a context-free grammar and S = fs0 , s1 , : : : , sg g be a linguistic term set, where V N denotes a set of nonterminal symbols, V T denotes a set of terminal symbols, I denotes the starting symbol, and P denotes the production rules. The elements of GH are defined as follows:

V N = fhprimary termi, hcomposite termi, hunary relationi, hbinary relationi, hconjunctionig; V T = flower than, greater than, at least, at most, between, and, s0 , s1 , : : : , sg g; I ∈ V N ; and P = fI :: =hprimary termijhcomposite termi hcomposite termi :: =hunary relationihprimary termijhbinary relationi hprimary termihconjunctionihprimary termi hprimary termi :: =s0 , s1 , : : : , sg hunary relationi :: =lower thanjgreater thanjat leastjat most hbinary relationi :: =between hconjunctioni :: =andg: Definition 3 Let EGH be a function that transforms the linguistic expressions ll obtained by the GH into a HFLTS H s of the linguistic term set S, i.e., EGH ∶ll → H s . The linguistic expressions gen-

erated by the production rules can be transformed into an HFLTS in different ways as follows: 1. 2. 3. 4. 5. 6.

EGH ðsi Þ = fsi jsi ∈ Sg; EGH ðlower than si Þ = fsj jsj ∈ S and sj < si g; EGH ðgreater than si Þ = fsj jsj ∈ S and sj > si g; EGH ðat least si Þ = fsj jsj ∈ S and sj ≥ si g; EGH ðat most si Þ = fsj jsj ∈ S and sj ≤ si g; and EGH ðbetween si and sj Þ = fsk jsk ∈ S and si ≤ sk ≤ sj g:



Definition 4 Let S = fs0 , s1 , : : : , sg g be a linguistic term set. The upper bound H S+ and the lower bound H s− of an HFLTS H s are, respectively, defined as follows:

1. H s+ = maxðsi Þ = sj , ∀i, si ≤ sj , si ∈ H s and sj ∈ H s , 2. H s− = minðsi Þ = sj , ∀i, si ≥ sj , si ∈ H s and sj ∈ H s . Definition 5 Let S = fs0 , s1 , : : : , sg g be a linguistic term set. The envelope of an HFLTS H s , i.e., envðH s Þ, is a linguistic interval that is obtained by the upper bound H s+ and the lower bound H s− of the HFLTS H s as

envðH s Þ = ½H s− , H s+ :

(1)

INTERVAL 2-TUPLE LINGUISTIC VARIABLES

The 2-tuple linguistic representation model was proposed by Herrera and Martinez [42] based on the concept of symbolic translation. As a generalization of the 2-tuple linguistic variables, Zhang [43] put forward an interval 2-tuple linguistic representation model for better modeling linguistic decision information. Definition 6 Let S = fs0 , s1 , : : : , sg g be a linguistic term set. An interval 2-tuple linguistic variable is composed of two 2-tuples, denoted by ½ðsk , α1 Þ, ðsl , α2 Þ, where ðsk , α1 Þ ≤ ðsl , α2 Þ. The interval 2-tuple that expresses the equivalent information to an interval value ½β1 , β2 ðβ1 , β2 ∈ ½0, 1, β1 ≤ β2 Þ is computed by [43,44]:

8 sk , k = roundðβ1 · gÞ, > > > s , l = roundðβ · gÞ, > > l 2 > < k Δ½β1 , β2 = Δ½ðsk , α1 Þ, ðsl , α2 Þ with α1 = β1 − g , α1 ∈ − 21g , > > > > > k > : α2 = β2 − g , α2 ∈ − 21g ,

1 2g 1 2g

,

(2)

:

On the contrary, an interval 2-tuple can be converted into an interval value ½β1 , β2 ðβ1 , β2 ∈ ½0, 1, β1 ≤ β2 Þ by using the function Δ−1 as follows:

Δ−1 ½ðsk , α1 Þ, ðsl , α2 Þ

k l = + α1 , + α2 = ½β1 , β2 : g g

(3)

It should be noted that the interval 2-tuple linguistic variable reduces to a 2-tuple linguistic variable when sk = sl and α1 = α2 . Definition 7 Let e a = ½ðr 1 , α1 Þ, ðt 1 , ε1 Þ and e b = ½ðr2 , α2 Þ, ðt 2 , ε2 Þ be two interval 2-tuples and λ ∈ ½0, 1, then the operation laws for the interval 2-tuples are defined as follows [13,45]:

1. e a+e b = ½ðr1 , α1 Þ, ðt 1 , ε1 Þ + ½ðr2 , α2 Þ, ðt 2 , ε2 Þ =Δ½Δ−1 ðr1 , α1 Þ + Δ−1 ðr 2 , α2 Þ, Δ−1 ðt 1 , ε1 Þ + Δ−1 ðt 2 , ε2 Þ; 2. e a×e b = ½ðr 1 , α1 Þ, ðt 1 , ε1 Þ × ½ðr2 , α2 Þ, ðt 2 , ε2 Þ =Δ½Δ−1 ðr1 , α1 Þ · Δ−1 ðr2 , α2 Þ, Δ−1 ðt 1 , ε1 Þ · Δ−1 ðt 2 , ε2 Þ;



3. λe a = λ½ðr1 , α1 Þ, ðt 1 , ε1 Þ = Δ½λΔ−1 ðr1 , α1 Þ, λΔ−1 ðt 1 , ε1 Þ; and 4. e aλ = ½ðr1 , α1 Þ, ðt 1 , ε1 Þλ = Δ½ðΔ−1 ðr 1 , α1 ÞÞλ , ðΔ−1 ðt 1 , ε1 ÞÞλ : Definition 8 Let e a = ½ðr1 , α1 Þ, ðt 1 , ε1 Þ and e b = ½ðr2 , α2 Þ, ðt 2 , ε2 Þ be two interval 2-tuples, then the degree of possibility of e a ≥ e b is defined as [13]:

δ2 − β 1 e ,0 ,0 , pðe a ≥ bÞ = max 1 − max hðe aÞ + hðe bÞ

(4)

where hðe aÞ = Δ−1 ðt 1 , ε1 Þ − Δ−1 ðr 1 , α1 Þ = δ1 − β1 and hðe bÞ = Δ−1 ðt 2 , ε2 Þ − Δ−1 ðr 2 , α2 Þ= Δ2 − β2 . To rank the interval 2-tuple linguistic arguments e ai = ½ðri , αi Þ, ðt i , εi Þði = 1, 2, : : : , nÞ, by using Eq 8, where p = pðe a≥e bÞ, we can construct a complementary matrix p = ½p ij

ij n×n

ij

pij ≥ 0, pij + pji = 1, pii = 0.5, i, j = 1, 2, : : : , n: Then, we can rank the arguments e ai ði = 1, 2, : : : , nÞ in accordance with the values of pi P pi = nj=1 pij ði = 1, 2, : : : , nÞ.

in descending order,

Definition 9 e = f½ðr 1 , α1 Þ, ðt 1 , ε1 Þ, : : : , ½ðrn , αn Þ, ðt n , εn Þg be a set of interval 2-tuples and Let X

w = ðw1 , w2 , : : : , wn ÞT be their associated weights with wj ∈ ½0, 1, j = 1, 2, : : : , n, Pn j=1 wj = 1. The interval 2-tuple weighted average (ITWA) operator is defined as follows [43,44]: e = ITWAf½ðr 1 , α1 Þ, ðt 1 , ε1 Þ, : : : , ½ðrn , αn Þ, ðt n , εn Þg ITWAðXÞ X n n X −1 −1 wi Δ ðri , αi Þ, wi Δ ðt i , εi Þ : =Δ i=1

(5)

i=1

Definition 10 Let e a = ½ðr1 , α1 Þ, ðt 1 , ε1 Þ and e b = ½ðr2 , α2 Þ, ðt 2 , ε2 Þ be two interval 2-tuples; then

Dðe a, e bÞ = Δ½maxðjΔ−1 ðr1 , α1 Þ − Δ−1 ðr 2 , α2 Þj + jΔ−1 ðt 1 , ε1 Þ − Δ−1 ðt 2 , ε2 ÞjÞ

(6)

is called the distance between e a and e b [38].

The Proposed Method for Robot Selection In this section, we focus on extending the classical MULTIMOORA to propose an HFL-MULTIMOORA method for robot evaluation and selection. The proposed model includes three main stages: (1) evaluating alternatives using HFLTSs and the interval 2-tuple linguistic method; (2) determining the importance coefficients of criteria by a combined weighting method; and (3) ranking the performances of alternative robots through an extended MULTIMOORA method. The flowchart of the proposed HFLMULTIMOORA model is presented in Fig. 1. The new method enables representing vague and uncertain qualitative data and solving robot selection problems taking into account both subjective and objective importance coefficients of evaluation criteria.



FIG. 1 Flowchart of the proposed robot selection method.

Suppose that a robot selection problem has l decision makers DMk ðk = 1, 2, : : : , lÞ; let Ai = fa1 , a2 , : : : , am g be a set of m alternatives and C j = fc1 , c2 , : : : , cn g be a set of n evaluation criteria. Each decision maker DMk is given a weight λk > 0 ðk = 1, 2, : : : , lÞ satisfying Pl k=1 λk = 1 to reflect his/her relative importance in the group decision-making process. When an expert is asked to evaluate the alternatives over each criterion, it is more convenient and straightforward for him/her to express performance assessments with linguistic expressions. Let Lk = ðllkij Þm×n be the linguistic decision matrix of the kth decision maker, where llkij is the linguistic information provided by DMk on the assessment of Ai with respect to C j . When all the linguistic expressions llkij ði = 1, 2, : : : , m; j = 1, 2, : : : , nÞ are determined, then the following linguistic judgment matrix can be 2 k ll11 llk12 6 llk21 llk22 6 Lk = 6 . .. 4 .. . llkm1 llkm1

developed: 3 · · · llk1n · · · llk2n 7 7 .. .. 7 . . 5 ···

(7)

llkmn

Let wk = ðwk1 , wk2 , : : : , wkn ÞT be the linguistic importance coefficient vector given by the kth decision maker, where wkj is the linguistic term assigned to C j by DMk . In what follows, the procedure for the HFL-MULTIMOORA method is explained: Step 1: Construct the hesitant fuzzy linguistic matrix. The linguistic judgment matrix Lk reflects the opinions of DMk on the alternatives over different criteria. However, we cannot directly compute with this matrix and have to transform it into a hesitant fuzzy linguistic matrix based on the transformation function EGH . After the transforming process, the hesitant fuzzy linguistic matrix H k is yielded as 2

H k11 6 H k21 6 Hk = 6 . 4 ..

H k12 H k22 .. .

··· ··· .. .

H k1n H k2n .. .

H km1

H km2

···

H kmn


3 7 7 7 5

(8)


where H kij = fskij jskij ∈ Sgði = 1, 2, : : : , n; j = 1, 2, : : : , nÞ and S = fs0 , s1 , : : : , sg g are a linguistic term set. Step 2: Obtain the envelope matrix Ek . According to Definition 5, the envelope for each HFLTS in the hesitant fuzzy linguistic matrix Hk can be obtained to construct the envelope matrix Ek . 2h

k+ H k− 11 , H 11

i

6h i 6 6 H k− , H k+ 21 6 21 Ek = envðH k Þ = 6 6 .. 6h 4 k− . k+ i H m1 , H m1

h i k+ H k− 11 , H 12 h i k+ H k− , H 22 22 .. . h i k+ H k− m2 , H m2

i3 h i7 7 k+ 7 k− H 2n , H 2n 7 7 7 .. . h i7 5 k+ H k− mn , H mn h

··· ··· .. . ···

k+ H k− 1n , H 1n

(9)

k k− k where H k+ ij is the upper bound of H ij and H ij means the lower bound of H ij .

ek . Step 3: Convert the envelope matrix Ek into the interval 2-tuple linguistic matrix R To deal with the hesitant fuzzy linguistic information provided by decision makers, we use the interval 2-tuple linguistic representation model because of its precision, simplicity, and interpretability in the computations with linguistic information [46]. Fig. 2 shows the way to obtain an HFLTS and convert it into interval 2-tuple linguistic terms. As ek = ðerkij Þ = a result, the interval 2-tuple linguistic matrix can be obtained as R m×n

ð½ðrkij , 0Þ, ðt kij , 0ÞÞm×n , where r kij , t kij ∈ S and r kij ≤ t kij .

Step 4: Aggregate decision makers’ opinions on constructing the collective interval ek = ðerkij Þ 2-tuple linguistic matrix R and the aggregated 2-tuple linguistic criteria imporm×n tance coefficients ðwj , αwj Þ, where i rij , αij , t ij , βij h i h i h i = ITWA r 1ij , 0 , t 1ij , 0 , r2ij , 0 , t 2ij , 0 , : : : , r lij , 0 , t lij , 0 ,

erij =

h

ðwj , αwj Þ = ITWAððw1j , 0Þ, ðw2j , 0Þ, : : : , ðwlj , 0ÞÞ X l =Δ λk Δ−1 ðwkj , 0Þ , j = 1, 2, : : : , n:

(10)

(11)

k=1

Step 5: Determine the subjective importance coefficients of criteria. Based on the aggregated 2-tuple linguistic importance coefficients of criteria ðwj , αwj Þ, j = 1, 2, : : : , n, the subjective importance coefficients of criteria wsj can be computed by Δ−1 ðwj , αwj Þ wsj = Pn , j = 1, 2, : : : , n: −1 j=1 Δ ðwj , αwj Þ

FIG. 2 The way to obtain an HFLTS and convert it to interval 2-tuples.


(12)


Step 6: Determine the objective importance coefficients of criteria. In this study, the concept of statistical variance [47] is used to determine the objective importance coefficients of criteria; the objective criteria importance coefficients woj are calculated by Δ−1 ε2j , j = 1, 2, : : : , n: woj = P n −1 ε2 j j=1 Δ

(13)

where X 1 m 2 ε2j = Δ ðΔ−1 dðer ij , rj* ÞÞ , j = 1, 2, : : : , n, m i=1 rj* =

Δð1Þ, for benefit criteria , j = 1, 2, : : : , n: Δð0Þ, for cost criteria

(14)

(15)

e 0 = ðerij0 Þ . Step 7: Compute the weighted interval 2-tuple linguistic matrix R m×n Considering the varying importance of the evaluation criteria, the weighted interval e 0 = ðerij0 Þ is computed by 2-tuple linguistic matrix R m×n er ij0

=

½ðrij0 , αij0 Þ, ðt ij0 , βij0 Þ

Δ−1 ðrij , αij Þ c Δ−1 ðt ij , βij Þ c = Δ wj , wj , q q

(16)

where wsj × woj wcj = Pn s o , j = 1, 2, : : : , n, j=1 wj × wj sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m X X q=1∕2 ðΔ−1 ðr ij , αij ÞÞ2 + ðΔ−1 ðt ij , βij ÞÞ2 : i=1

(17)

(18)

i=1

Step 8: The HFL ratio system. For optimization, the assessments of decision makers are added in cases of maximization and subtracted in cases of minimization for each alternative. Hence, the overall assessments of alternatives Ai with respect to every criterion e yi are calculated by e yi =

g X j=1

erij0 −

n X

erij0 ,

(19)

j=g+1

where Cj for j = 1,2,…,g are the benefit criteria and Cj for j = g + 1, g + 2, …, n are the cost criteria. Step 9: The HFL reference point approach. e 0 , the 2-tuple maximal obBased on the weighted interval 2-tuple linguistic matrix R + + jective reference point (MORP) can be defined as ðr j , αj Þ = maxi fðt ij , βij Þg and ðr+j , α+j Þ = mini fðr ij , αij Þg for the benefit criteria and cost criteria, respectively. Then, the distance matrix D = ½d ij m×n is acquired by



d ij = dðer ij0 , ðr+j , α+j ÞÞ = Δ½maxðjΔ−1 ðr ij0 , αij0 Þ − Δ−1 ðr+j , α+j Þj, jΔ−1 ðt ij0 , βij0 Þ − Δ−1 ðr+j , α+j ÞjÞ,

(20)

where the distance d ij shows the gap of the alternative Ai with respect to the criterion C j . The distance of each alternative from the 2-tuple MORP can be computed by di = max d ij : j

(21)

Step 10: The HFL full multiplicative form. The overall utility of the ith alternative is expressed as an interval 2-tuple by using Eq 26. AQ1 e ui =

e ai , e bi

(22)

Qg Q where e a = j=1 erij0 denotes the product of the benefit criteria and e bi = nj=g+1 er ij0 is the product of the cost criteria. Step 11: Rank the robot alternatives. Finally, three ranking lists of the considered robots can be obtained by sorting the ui for i = 1,2,…,m in descending order and the 2-tuple distances di interval 2-tuples e yi and e for i = 1,2,…,m in increasing order. Then, the final ranking order of all the alternatives is determined based on the dominance theory [48].

An Illustrative Example CASE ILLUSTRATION

In this section, a robot selection problem of an auto company [7] is considered to illustrate the computation and applicability of the proposed HFL-MULTIMOORA method. After preliminary simulation, it has been found that there are eight robots (R1 , R2 , : : : , R8 ) satisfying the requirements of this particular problem. A committee of three decision makers ðDM1 , DM2 , and DM3 Þ was created in order to evaluate and select the most appropriate robot for this application. The members of this team are the production managers of the factory production line. The criteria C j ðj = 1, 2, : : : , 7Þ, which have been considered for the analysis, are as follows: C 1 = Inconsistency with infrastructure; C 2 = Man-machine interface; C 3 = Programming flexibility; C 4 = Vendor’s service contract; C 5 = Supporting channel partner’s performance; C 6 = Compliance; and C 7 = Stability. All three decision makers provide their assessments in a set of seven labels, A. In addition, the relative importance of the seven criteria was rated by the decision makers with a set of five linguistic labels, B. These linguistic term sets are denoted as follows: A = fa0 = Very poor ðVPÞ, a1 = Poor ðPÞ, a2 = Moderately poor ðMPÞ, a3 = Fair ðFÞ, a4 = Moderately good ðMGÞ, a5 = Good ðGÞ, a6 = Very good ðVGÞg:



B = fb0 = Very low ðVLÞ, b1 = Low ðLÞ, b2 = Medium ðMÞ, b3 = High ðHÞ, b4 = Very high ðVHÞg: With these linguistic term sets and the context-free grammar, the linguistic evaluations on the alternative robots provided by the three decision makers are shown in Tables 1–3. The importance coefficients of the criteria determined are shown in Table 4. TABLE 1 Linguistic assessments of robots provided by the first decision maker. DM1 R1 R2

C1

C2

C3

C4

C5

Between VL and L Between H and VH Between ML and MH Between MH and H Between MH and H Greater than VH

At most ML

C6

C7

At least H

Greater than VH

M

Greater than MH

MH

Between ML and MH

M

Between ML and M

Between L and ML

Between M and H

Between MH and VH Between ML and M

Between M and H

Between L and ML

R3

H

R4

ML

Between ML and M

At least MH

R5

Greater than VH

Between VL and L

Between L and ML

Between VL and L

Between L and ML Between VL and ML

Between L and ML Between ML and MH

M ML

R6

L

Between M and H

Between M and H

Between ML and M

At least VH

Greater than MH

R7

At least MH

Between L and M

At most L

ML

Between L and ML

At most VL

ML

R8

Greater than VH

L

Between VL and L

Between VL and L

L

Between L and M

Between ML and M

C5

C6

C7

At least H

Between MH and VH

H

Greater than MH

M

Between L and ML

M

Between VL and L

Between MH and VH

ML

Between M and MH

Between ML and M

ML

At most L

Between H and VH

TABLE 2 Linguistic assessments of robots provided by the second decision maker. DM2

C1

C2

C3

C4

R1

Between VL and L

Greater than VH

R2

Greater than VH

Between L and M

At most L

Between L and M

R3

Between MH and H

At most L

Between ML and M

Between L and M

R4

At most L

R5

Between MH and H Between MH and H

Between ML and M Between M and MH Between M and MH

Between MH and H Between L and ML

ML

Between VL and L Between L and ML

R6

At most VL

Greater than H

Between H and VH

Between L and M

R7

Greater than VH

Between ML and MH

At most ML

M

R8

H

VL

Between VL and ML Between VL and ML

Greater than VH Between MH and VH Between ML and M Between L and ML At most VL

M

VH At most L Between ML and M

TABLE 3 Linguistic assessments of robots provided by the third decision maker. DM3

C1

C2

C3

C4

C5

Between ML and MH Between H and VH Between MH and H

C6

R1

Between VL and L

Greater than VH

R2

Between MH and H

Between L and M

Between L and M

R3

Between MH and H

At most L

Between VL and ML

M

R4

ML

Between ML and M

Between MH and H

Greater than H

R5

Greater than VH

At most L

Between L and ML

Between VL and L Between ML and M

Between VL and L Between MH and H Between M and MH

M

C7

At least MH

VH

Greater than H

Between ML and MH

M

Between L and ML

Between M and H

ML

Between M and MH

M

ML

ML

Between M and MH

R6

L

Between M and H

Between M and H

R7

Greater than VH

Between MH and H

ML

M

Between L and ML

VL

ML

R8

Greater than VH

VL

Between ML and M

VL

L

Between L and M

At most L


Greater than H


TABLE 4 Criteria importance coefficients evaluated by the decision makers. Decision Makers Criteria

DM1

DM2

DM3

C1

H

VH

VH

C2

M

M

MH

C3

H

MH

H

C4

MH

VH

VH MH

C5

H

MH

C6

MH

H

H

C7

H

MH

MH

In the following, we utilize the HFL-MULIMOORA model to determine the most preferred robot, which includes the following steps: Step 1: Based on the linguistic expressions in Tables 1–3 and using the transformation function EGH , we can generate the following hesitant fuzzy linguistic matrixes: 2

fa5 ,a6 g fa0 ,a1 g 6 fa6 g fa 0 ,a1 ,a2 g 6 6 fa5 g fa1 ,a2 g 6 6 fa2 g fa2 ,a3 g H1 = 6 6 fa6 g fa0 ,a1 g 6 6 fa1 g fa 3 ,a4 ,a5 g 6 4 fa4 ,a5 ,a6 g fa1 ,a2 ,a3 g fa1 g fa6 g

fa2 ,a3 ,a4 g fa4 ,a5 g fa4 ,a5 g fa1 ,a2 g fa3 g fa4 ,a5 ,a6 g fa0 ,a1 ,a2 g fa3 g fa2 ,a3 g fa4 ,a5 ,a6 g fa4 ,a5 ,a6 g fa2 ,a3 g fa2 ,a3 g fa0 ,a1 g fa1 ,a2 g fa3 ,a4 ,a5 g fa2 ,a3 g fa6 g fa0 ,a1 g fa2 g fa1 ,a2 g fa0 ,a1 g fa0 ,a1 g fa1 g

3 fa5 ,a6 g fa6 g fa4 g fa2 ,a3 ,a4 g 7 7 fa1 ,a2 g fa3 ,a4 ,a5 g 7 7 fa3 ,a4 ,a5 g fa3 g 7 7 fa2 ,a3 ,a4 g fa2 g 7 7 fa4 ,a5 ,a6 g fa5 ,a6 g 7 7 fa0 g fa2 g 5 fa1 ,a2 ,a3 g fa2 ,a3 g

2

3 fa6 g fa4 ,a5 g fa4 ,a5 g fa5 ,a6 g fa4 ,a5 ,a6 g fa5 g fa0 ,a1 g 6 fa6 g fa1 ,a2 ,a3 g fa0 ,a1 g fa1 ,a2 ,a3 g fa4 ,a5 ,a6 g fa3 g fa1 ,a2 g 7 6 7 6 fa4 ,a5 g fa0 ,a1 g 7 fa ,a g fa ,a g fa g fa ,a g fa 2 3 2 3 3 0 1 4 ,a5 ,a6 g 7 6 6 fa0 ,a1 g fa2 ,a3 g 7 fa ,a g fa ,a g fa g fa ,a g fa ,a g 3 4 3 4 2 3 4 2 3 7 H2 = 6 6 fa4 ,a5 g fa1 ,a2 g fa2 g fa0 g fa1 ,a2 g fa2 g fa0 ,a1 g 7 6 7 6 fa0 g fa5 ,a6 g fa5 ,a6 g fa1 ,a2 ,a3 g fa6 g fa4 ,a5 ,a6 g fa6 g 7 6 7 4 fa6 g fa2 ,a3 ,a4 g fa0 ,a1 ,a2 g fa3 g fa2 ,a3 g fa1 ,a2 g fa0 ,a1 g 5 fa0 g fa0 ,a1 ,a2 g fa0 ,a1 ,a2 g fa0 g fa3 g fa2 ,a3 g fa5 g

2

fa6 g fa0 , a1 g 6 fa4 , a5 g fa1 , a2 , a3 g 6 6 fa4 , a5 g fa0 , a1 g 6 6 fa2 g fa2 , a3 g H3 = 6 6 fa6 g fa0 , a1 g 6 6 fa1 g fa3 , a4 , a5 g 6 4 fa6 g fa4 , a5 g fa0 g fa6 g

fa2 , a3 , a4 g fa5 , a6 g fa1 , a2 g fa3 g fa0 , a1 , a2 g fa3 g fa4 , a5 , a6 g fa4 , a5 , a6 g fa1 , a2 g fa0 , a1 g fa3 , a4 , a5 g fa0 , a1 g fa2 g fa3 g fa2 , a3 g fa0 g


3 fa4 , a5 g fa4 , a5 , a6 g fa6 g fa3 , a4 g fa5 , a6 g fa2 , a3 , a4 g 7 7 fa3 g fa1 , a2 g fa3 , a4 , a5 g 7 7 fa2 g fa3 , a4 g fa3 g 7 7 fa2 , a3 g fa2 g fa2 g 7 7 fa4 , a5 g fa3 , a4 g fa5 , a6 g 7 7 fa1 , a2 g fa0 g fa2 g 5 fa1 g fa1 , a2 , a3 g fa0 , a1 g


Step 2: According to the computing model defined for HFLTSs, we can acquire the following envelope matrixes: 2

½a0 , a1 6 ½a6 , a6 6 6 ½a5 , a5 6 6 ½a2 , a2 E1 = 6 6 ½a6 , a6 6 6 ½a1 , a1 6 4 ½a4 , a6 ½a6 , a6 2

½a0 , a1 6 ½a6 , a6 6 6 ½a4 , a5 6 6 ½a0 , a1 E2 = 6 6 ½a4 , a5 6 6 ½a0 , a0 6 4 ½a6 , a6 ½a5 , a5 2

½a0 , a1 6 ½a4 , a5 6 6 ½a4 , a5 6 6 ½a2 , a2 E3 = 6 6 ½a6 , a6 6 6 ½a1 , a1 6 4 ½a6 , a6 ½a6 , a6

½a5 , a6 ½a0 , a2 ½a1 , a2 ½a2 , a3 ½a0 , a1 ½a3 , a5 ½a1 , a3 ½a1 , a1





3 ½a6 , a6 ½a2 , a4 7 7 ½a3 , a5 7 7 ½a3 , a3 7 7 ½a2 , a2 7 7 ½a5 , a6 7 7 ½a2 , a2 5 ½a2 , a3













Step 3: Convert the interval linguistic evaluations into interval 2-tuples and 2-tuples, respectively. The results are shown in Tables 5–6. Step 4: The aggregated linguistic ratings of alternatives and the aggregated importance coefficients of criteria are calculated to determine the collective interval 2-tuple linguistic matrix and the 2-tuple linguistic criteria importance coefficients (see Table 7). Steps 5 and 6: Utilizing Eqs 12–15, the subjective and objective importance coefficients of the seven evaluation criteria are computed as given in Table 8. Note that the combined importance coefficients can be derived by using Eq 17 as also listed in Table 8. e 0 = ðer ij0 Þ is computed by Step 7: The weighted interval 2-tuple linguistic matrix R m×n employing Eqs 16–18 and presented in Table 9. Steps 8–10: The assessments of the three decision makers are aggregated by Eq 19 to obtain the overall assessment of each alternative e yi . Next, the distance from the 2-tuple MORP vector dij is calculated by using Eq 20 for each of the robot alternatives, and the overall utilities of the eight alternatives e ui are computed by Eq 22. Table 10 summarizes the results obtained by using the ratio system, the reference point approach, and the full multiplicative form. Step 11: The alternatives are prioritized in accordance with the ranking indexes e yi , d ij and e ui referring to Definition 8. Then, the theory of dominance is used to aggregate the three rank lists into a single final rank. The obtained ranking lists of the eight robots are shown in Table 11.



TABLE 5 Interval 2-tuple linguistic matrixes of the three decision makers. Criteria Decision Makers DM1

DM2

DM3

Alternatives

C1

C2

C3

C4

C5

C6

C7

R1

[(a0,0), (a1,0)]

[(a5,0), (a6,0)]

[(a2,0), (a4,0)]

[(a4,0), (a5,0)]

[(a4,0), (a5,0)]

[(a5,0), (a6,0)]

[(a6,0), (a6,0)]

R2

[(a6,0), (a6,0)]

[(a0,0), (a2,0)]

[(a1,0), (a2,0)]

[(a3,0), (a3,0)]

[(a4,0), (a6,0)]

[(a4,0), (a4,0)]

[(a2,0), (a4,0)]

R3

[(a5,0), (a5,0)]

[(a1,0), (a2,0)]

[(a0,0), (a2,0)]

[(a3,0), (a3,0)]

[(a2,0), (a3,0)]

[(a1,0), (a2,0)]

[(a3,0), (a5,0)]

R4

[(a2,0), (a2,0)]

[(a2,0), (a3,0)]

[(a4,0), (a6,0)]

[(a4,0), (a6,0)]

[(a2,0), (a3,0)]

[(a3,0), (a5,0)]

[(a3,0), (a3,0)]

R5

[(a6,0), (a6,0)]

[(a0,0), (a1,0)]

[(a1,0), (a2,0)]

[(a0,0), (a1,0)]

[(a1,0), (a2,0)]

[(a2,0), (a4,0)]

[(a2,0), (a2,0)]

R6

[(a1,0), (a1,0)]

[(a3,0), (a5,0)]

[(a3,0), (a5,0)]

[(a2,0), (a3,0)]

[(a6,0), (a6,0)]

[(a4,0), (a6,0)]

[(a5,0), (a6,0)]

R7

[(a4,0), (a6,0)]

[(a1,0), (a3,0)]

[(a0,0), (a1,0)]

[(a2,0), (a2,0)]

[(a1,0), (a2,0)]

[(a0,0), (a0,0)]

[(a2,0), (a2,0)]

R8

[(a6,0), (a6,0)]

[(a1,0), (a1,0)]

[(a0,0), (a1,0)]

[(a0,0), (a1,0)]

[(a1,0), (a1,0)]

[(a1,0), (a2,0)]

[(a2,0), (a3,0)]

R1

[(a0,0), (a1,0)]

[(a6,0), (a6,0)]

[(a4,0), (a5,0)]

[(a4,0), (a5,0)]

[(a5,0), (a6,0)]

[(a4,0), (a6,0)]

[(a5,0), (a5,0)]

R2

[(a6,0), (a6,0)]

[(a1,0), (a3,0)]

[(a0,0), (a1,0)]

[(a1,0), (a3,0)]

[(a4,0), (a6,0)]

[(a3,0), (a3,0)]

[(a1,0), (a2,0)]

R3

[(a4,0), (a5,0)]

[(a0,0), (a1,0)]

[(a2,0), (a3,0)]

[(a2,0), (a3,0)]

[(a3,0), (a3,0)]

[(a0,0), (a1,0)]

[(a4,0), (a6,0)]

R4

[(a0,0), (a1,0)]

[(a2,0), (a3,0)]

[(a3,0), (a4,0)]

[(a3,0), (a4,0)]

[(a2,0), (a2,0)]

[(a3,0), (a4,0)]

[(a2,0), (a3,0)]

R5

[(a4,0), (a5,0)]

[(a1,0), (a2,0)]

[(a2,0), (a2,0)]

[(a0,0), (a1,0)]

[(a1,0), (a2,0)]

[(a2,0), (a2,0)]

[(a0,0), (a1,0)]

R6

[(a0,0), (a0,0)]

[(a5,0), (a6,0)]

[(a5,0), (a6,0)]

[(a1,0), (a3,0)]

[(a6,0), (a6,0)]

[(a4,0), (a6,0)]

[(a6,0), (a6,0)]

R7

[(a6,0), (a6,0)]

[(a2,0), (a4,0)]

[(a0,0), (a2,0)]

[(a3,0), (a3,0)]

[(a2,0), (a3,0)]

[(a1,0), (a2,0)]

[(a0,0), (a1,0)]

R8

[(a5,0), (a5,0)]

[(a0,0), (a0,0)]

[(a0,0), (a2,0)]

[(a0,0), (a2,0)]

[(a0,0), (a0,0)]

[(a3,0), (a3,0)]

[(a2,0), (a3,0)]

R1

[(a0,0), (a1,0)]

[(a6,0), (a6,0)]

[(a2,0), (a4,0)]

[(a5,0), (a6,0)]

[(a4,0), (a5,0)]

[(a4,0), (a6,0)]

[(a6,0), (a6,0)]

R2

[(a3,0), (a5,0)]

[(a1,0), (a3,0)]

[(a1,0), (a2,0)]

[(a3,0), (a3,0)]

[(a3,0), (a4,0)]

[(a5,0), (a6,0)]

[(a2,0), (a4,0)]

R3

[(a3,0), (a5,0)]

[(a0,0), (a1,0)]

[(a0,0), (a2,0)]

[(a3,0), (a3,0)]

[(a3,0), (a3,0)]

[(a1,0), (a2,0)]

[(a3,0), (a5,0)]

R4

[(a2,0), (a2,0)]

[(a2,0), (a3,0)]

[(a4,0), (a6,0)]

[(a4,0), (a6,0)]

[(a2,0), (a2,0)]

[(a3,0), (a4,0)]

[(a3,0), (a3,0)]

R5

[(a6,0), (a6,0)]

[(a0,0), (a1,0)]

[(a1,0), (a2,0)]

[(a0,0), (a1,0)]

[(a2,0), (a3,0)]

[(a2,0), (a2,0)]

[(a2,0), (a2,0)]

R6

[(a1,0), (a1,0)]

[(a3,0), (a5,0)]

[(a3,0), (a5,0)]

[(a0,0), (a1,0)]

[(a4,0), (a5,0)]

[(a3,0), (a4,0)]

[(a5,0), (a6,0)]

R7

[(a6,0), (a6,0)]

[(a4,0), (a5,0)]

[(a2,0), (a2,0)]

[(a3,0), (a3,0)]

[(a1,0), (a2,0)]

[(a0,0), (a0,0)]

[(a2,0), (a2,0)]

R8

[(a6,0), (a6,0)]

[(a0,0), (a0,0)]

[(a2,0), (a3,0)]

[(a0,0), (a0,0)]

[(a1,0), (a1,0)]

[(a1,0), (a3,0)]

[(a0,0), (a1,0)]

TABLE 6 The 2-Tuple linguistic criteria importance coefficients. Decision Makers Criteria

DM1

DM2

DM3

C1

(b5,0)

(b6,0)

(b6,0)

C2

(b3,0)

(b3,0)

(b4,0)

C3

(b5,0)

(b4,0)

(b5,0)

C4

(b4,0)

(b6,0)

(b6,0)

C5

(b5,0)

(b4,0)

(b4,0)

C6

(b4,0)

(b5,0)

(b5,0)

C7

(b5,0)

(b4,0)

(b4,0)

As can be seen from Table 11, the final ranking of the eight alternatives is R1 > R6 > R4 > R2 > R3 > R7 > R5 > R8 according to the proposed HFL-MULTIMOORA method. Therefore, the most suitable robot is R1 for the given application. COMPARATIVE ANALYSIS AND DISCUSSIONS

To illustrate the advantages of our proposed HFL-MULTIMOORA, a comparative analysis is conducted with some previous robot selection models, which include the interval



TABLE 7 Collective interval 2-tuple linguistic matrix and the aggregated 2-tuple linguistic importance coefficients of criteria. Criteria Alternatives

C1

C2

C3

C4

C5

C6

C7

R1

Δ[0.000,0.167]

Δ[0.944,1.000]

Δ[0.444,0.722]

Δ[0.722,0.889]

Δ[0.722,0.889]

Δ[0.722,1.000]

Δ[0.944,0.944]

R2

Δ[0.833,0.944]

Δ[0.111,0.444]

Δ[0.111,0.278]

Δ[0.389,0.500]

Δ[0.611,0.889]

Δ[0.667,0.722]

Δ[0.278,0.556]

R3

Δ[0.667,0.833]

Δ[0.056,0.222]

Δ[0.111,0.389]

Δ[0.444,0.500]

Δ[0.444,0.500]

Δ[0.111,0.278]

Δ[0.556,0.889]

R4

Δ[0.222,0.278]

Δ[0.056,0.222]

Δ[0.611,0.889]

Δ[0.611,0.889]

Δ[0.333,0.389]

Δ[0.500,0.722]

Δ[0.444,0.500]

R5

Δ[0.889,0.944]

Δ[0.333,0.500]

Δ[0.222,0.333]

Δ[0.000,0.167]

Δ[0.222,0.389]

Δ[0.333,0.444]

Δ[0.222,0.278]

R6

Δ[0.111,0.111]

Δ[0.056,0.222]

Δ[0.611,0.889]

Δ[0.167,0.389]

Δ[0.889,0.944]

Δ[0.611,0.889]

Δ[0.889,1.000]

R7

Δ[0.899,1.000]

Δ[0.389,0.667]

Δ[0.111,0.278]

Δ[0.444,0.444]

Δ[0.222,0.389]

Δ[0.056,0.111]

Δ[0.222,0.278]

R8

Δ[0.944,0.944]

Δ[0.056,0.056]

Δ[0.111,0.333]

Δ[0.000,0.167]

Δ[0.111,0.111]

Δ[0.278,0.444]

Δ[0.222,0.389]

Δ(0.944)

Δ(0.556)

Δ(0.778)

Δ(0.889)

Δ(0.722)

Δ(0.778)

Δ(0.722)

Weights

TABLE 8 Criteria importance coefficients by the subjective, objective, and combined weighting methods. C1

C2

C3

C4

C5

C6

C7

ws

0.175

0.103

0.144

0.165

0.134

0.144

0.134

wo

0.308

0.177

0.080

0.146

0.110

0.074

0.105

wc

0.366

0.124

0.079

0.163

0.100

0.072

0.096

Note: ws = subjective weighting method; wo = objective weighting method; and wc = combined weighting method.

TABLE 9 Normalized interval 2-tuple linguistic matrix and the 2-tuple MORP vector. Criteria Alternatives

C1

C2

C3

C4

C5

C6

C7

R1

Δ[0.000, 0.043]

Δ[0.113, 0.119]

Δ[0.037, 0.060]

Δ[0.118, 0.146]

Δ[0.063, 0.077]

Δ[0.046, 0.064]

Δ[0.074, 0.074]

R2

Δ[0.214, 0.242]

Δ[0.013, 0.053]

Δ[0.009, 0.023]

Δ[0.064, 0.082]

Δ[0.053, 0.077]

Δ[0.042, 0.046]

Δ[0.022, 0.044]

R3

Δ[0.171, 0.214]

Δ[0.007, 0.026]

Δ[0.009, 0.032]

Δ[0.073, 0.082]

Δ[0.039, 0.043]

Δ[0.007, 0.018]

Δ[0.044, 0.070]

R4

Δ[0.057, 0.071]

Δ[0.040, 0.060]

Δ[0.050, 0.073]

Δ[0.100, 0.146]

Δ[0.029, 0.034]

Δ[0.032, 0.046]

Δ[0.035, 0.039]

R5

Δ[0.228, 0.242]

Δ[0.007, 0.026]

Δ[0.018, 0.027]

Δ[0.000, 0.027]

Δ[0.019, 0.034]

Δ[0.021, 0.028]

Δ[0.017, 0.022]

R6

Δ[0.028, 0.028]

Δ[0.073, 0.106]

Δ[0.050, 0.073]

Δ[0.027, 0.064]

Δ[0.077, 0.082]

Δ[0.039, 0.057]

Δ[0.070, 0.079]

R7

Δ[0.228, 0.256]

Δ[0.046, 0.079]

Δ[0.009, 0.023]

Δ[0.073, 0.073]

Δ[0.019, 0.034]

Δ[0.004, 0.007]

Δ[0.017, 0.022]

R8

Δ[0.242, 0.242]

Δ[0.007, 0.007]

Δ[0.009, 0.027]

Δ[0.000, 0.027]

Δ[0.010, 0.010]

Δ[0.018, 0.028]

Δ[0.017, 0.031]

Δ(0.256)

Δ(0.119)

Δ(0.073)

Δ(0.146)

Δ(0.082)

Δ(0.064)

Δ(0.079)

(r+, α+)

TABLE 10 The ranking indexes e y i , di, and e ui for the eight alternatives. Alternatives

e yi

di

e ui

R1

Δ[0.451, 0.582]

Δ(0.128)

Δ[0.000, 1.61E-08]

R2

Δ[0.417, 0.567]

Δ(0.114)

Δ[8.14E-11, 3.74E-09]

R3

Δ[0.349, 0.485]

Δ(0.086)

Δ[9.02E-12, 7.99E-10]

R4

Δ[0.343, 0.469]

Δ(0.073)

Δ[3.68E-10, 2.77E-09]

R5

Δ[0.311, 0.407]

Δ(0.114)

Δ[0.000, 1.01E-10]

R6

Δ[0.365, 0.489]

Δ(0.100)

Δ[6.00E-10, 5.15E-09]

R7

Δ[0.396, 0.494]

Δ(0.128)

Δ[8.42E-12, 1.78E-10]

R8

Δ[0.303, 0.372]

Δ(0.114)

Δ[0.000, 1.01E-11]



TABLE 11 Rankings of the alternatives according to the HFL-MULTIMOORA. Alternatives

e yi

di

e ui

Final Ranking

R1

1

1

1

1

R2

2

7

2

4

R3

5

6

5

5

R4

6

3

4

3

R5

7

4

7

7

R6

4

2

3

2

R7

3

8

6

6

R8

8

4

8

8

type-2 fuzzy VIKOR (ITF-VIKOR) [7], the interval 2-tuple linguistic TOPSIS (ITLTOPSIS) [13], the interval-valued fuzzy TOPSIS (IVF-TOPSIS) [28], and the interval 2-tuple linguistic VIKOR (ITL-VIKOR) [49]. Fig. 3 displays the ranking results of the eight alternative robots as yielded using the listed methods. From Fig. 3, we can observe that all the five methods suggest robot R1 as the first choice for the considered industrial application. Moreover, the ranking orders of the eight robots determined by the proposed method are exactly matched with those derived by the ITL-TOPSIS and the IVF-TOPSIS methods. This demonstrates the validity of the proposed decision-making framework. On the other hand, there are some differences between the ranking obtained by the proposed HFL-MULTIMOORA and those determined with the ITF-VIKOR (for R2 and R3) and the ITL-VIKOR (for R2 and R5) approaches. The main reasons for these differences may be as follows: first, the ITF-VIKOR uses interval type-2 fuzzy sets and the ITL-VIKOR employs interval 2-tuples to deal with the ambiguity and uncertainty in evaluating the robots. However, the interval type-2 fuzzy method led to a certain degree

FIG. 3 Comparative ranking of the alternative robots.



of information loss in the linguistic information transformation process. The interval 2-tuple linguistic method can avoid information distortion and loss; but it is unable to express the real preferences of decision makers and reflect their uncertainty, hesitancy, and inconsistency. In contrast, HFLTSs are combined with interval 2-tuple linguistic variables in this study for robot assessments, which is especially useful in handling qualitative robot selection problems when the evaluation values of alternatives are represented as linguistic expressions. Second, the VIKOR algorithm is adopted by the ITF-VIKOR and the ITL-VIKOR methods to determine the priority orders of robots. These two methods only analyze the alternative robots from a single angle. By comparison, the MULTIMOORA utilized in our proposed method is more comprehensive in dealing with robot selection problems as it utilizes the ratio system, the reference point approach, and the full multiplicative form. Therefore, the reliability and veracity of the decision-making results would be improved greatly by using the proposed HFL-MULTIMOORA model. Compared with other robot evaluation and selection methods in the literature, the HFL-MULTIMOORA method developed in this study has the following attractions: 1. The HFLTSs improve the elicitation of linguistic information when a decision maker hesitates among several values to assess a robot. The use of HFLTSs and interval 2-tuples provides a more flexible way to represent decision makers’ evaluations. So, an organized method is given to combine expert knowledge and experience for use in selecting the optimal robot for a manufacturing company. 2. Both subjective and objective importance coefficients of criteria are taken into account in the process of robot evaluation and selection, which makes the proposed decision-making model more realistic, more practical, and more flexible. 3. The MULTIMOORA method is used for the prioritization of robot alternatives, which is a robust and powerful MCDM method and is easily implemented relative to other methods such as the TOPSIS and the VIKOR methods. Hence, the proposed method more effectively conducts robust evaluation for a particular manufacturing environment.

Conclusions In this article, an extended MULTIMOORA method with HFLTSs and interval 2-tuples was proposed to find the best robot for a specific robot selection problem. We considered both subjective and objective importance coefficients of criteria in the process of prioritization of robots. In addition, a numerical example has been provided to demonstrate the applicability and effectiveness of our proposed decision-making method. The results derived using the HFL-MULTIMOORA are relatively consistent with those obtained by past researchers. Furthermore, it is proved that the extended MULTIMOORA method offers a useful, practical, and flexible way to solve the robot selection problem under hesitant and uncertain information environment. In future research, the following directions are recommended. First, this study employs a robot selection example with eight alternatives and seven commonly used criteria to demonstrate the proposed integrated linguistic MCDM framework. In the future, actual applications of this new methodology to more complex case studies are suggested to further show its robustness and efficiency. Second, a problem may arise for the proposed HFL-MULTIMOORA where an importance coefficient does not influence the ranking outcome. Hence, it would be better to introduce subcriteria for criteria in order to give more importance to an objective in future research. Additionally, the proposed model



based on HFLTSs and the MULTIMOORA method requires many computations to obtain the ideal solution for a particular industrial application. Thus, a computer-based application system can be developed for the further work to quicken the implementation of the proposed robot selection method and facilitate a man-machine interaction presentation of the selection and analysis results. ACKNOWLEDGMENTS

The authors are very grateful to the editor and reviewers for their insightful and constructive comments and suggestions, which are very helpful in improving the quality of the article. This work was partially supported by the National Natural Science Foundation of China (Nos. 61773250, 71671125, and 71402090), the Shanghai Pujiang Talents Program (No. 15PJC050), and the Program for Shanghai Young Eastern Scholar (No. QD2015019).

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