An integrated fuzzy MCDM based approach for robot selection considering objective and subjective criteria

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

An integrated fuzzy MCDM based approach for robot selection considering objective and subjective criteria

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R. Parameshwaran ∗ , S. Praveen Kumar, K. Saravanakumar Department of Mechatronics Engineering, Kongu Engineering College, Perundurai 638052, Tamilnadu, India

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a r t i c l e

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a b s t r a c t

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Article history: Received 25 July 2013 Received in revised form 14 July 2014 Accepted 18 September 2014 Available online xxx

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Keywords: Robot selection FDM FAHP Fuzzy TOPSIS Fuzzy VIKOR and Brown–Gibson model

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1. Introduction

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22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

Robots with vastly different capabilities and specifications are available for a wide range of applications. Selection of a robot for a specific application has become more complicated due to increase in the complexity, advanced features and facilities that are continuously being incorporated into the robots by different manufacturers. The aim of this paper is to present an integrated approach for the optimal selection of robots by considering both objective and subjective criteria. The approach utilizes Fuzzy Delphi Method (FDM), Fuzzy Analytical Hierarchical Process (FAHP), Fuzzy modified TOPSIS or Fuzzy VIKOR and Brown–Gibson model for robot selection. FDM is used to select the list of important objective and subjective criteria based on the decision makers’ opinion. Fuzzy AHP method is then used to find out the weight of each criterion (both objective and subjective). Fuzzy modified TOPSIS or Fuzzy VIKOR method is then used to rank the alternatives based on objective and subjective factors. The rankings obtained are used to calculate the robot selection index based on Brown–Gibson model. The proposed methodology is illustrated with a case study related to selection of robot for teaching purpose. It is found that the highest ranked alternative based on Fuzzy VIKOR is closest to the ideal solution. © 2014 Elsevier B.V. All rights reserved.

Robots are very powerful elements of today’s industry and are defined as automatically controlled, reprogrammable, multipurpose manipulators programmable in three or more axes [1]. The recent advancement in automation field has lead to increased usage of robots with distinct capabilities, features and specifications. A company’s competitiveness in terms of the productivity of its facilities and quality of its products will be adversely affected by improper selection of robots [2,3]. The determination of the most appropriate robot considering multiple conflicting qualitative and quantitative criteria has been a difficult task for the decision makers. Literature reveals that multi-criteria decision making (MCDM) methods, production system performance optimization models, computer-assisted models and a general category of solutions are employed in robot selection process [4–6]. Among these models, MCDM is the most common method for ranking robots and production system performance optimization models are rarely used [5,6]. Hence, the methodologies for robot selection are divided into

∗ Corresponding author. Tel.: +91 9865919915. E-mail addresses: paramesh [email protected], [email protected], parameshwaran [email protected] (R. Parameshwaran), [email protected] (S. Praveen Kumar), [email protected] (K. Saravanakumar).

three broad categories: (a) MCDM, (b) integrated approaches, and (c) a general category of solutions. Various MCDM methods reported in the literature for robot selection process are: AHP (Analytic Hierarchy Process), TOPSIS (technique for order preference by similarity to ideal solution), VIKOR (VIsekriterijumsko KOmpromisno Rangiranje), ELECTRE II (ELimination and Et Choice Translating REality), PROMETHEE II (Preference Ranking Organization METHod for Enrichment Evaluation) and DEA (Data Envelopment Analysis). Due to vagueness in the data and ambiguity in decision-making process, fuzzy set theory has been incorporated into MCDM techniques by many researchers for robot selection problem. The selection criteria and the techniques considered by various researchers are presented in Table 1. Among these techniques, TOPSIS and VIKOR methods seemed to be more appropriate for solving the robot selection problem because they have capability to deal with each kind of judgment criteria, having clarity of results and easiness to deal with attributes and decision options [12]. In the integrated approach category, one or more techniques are integrated or combined to select robot for various applications. Table 2 provides the list of works related to integrated approach category along with their selection criteria. The general category of robot selection models includes statistical models, mathematical models and soft computing based models. Table 3 provides the list of works related to general solution category along with their

http://dx.doi.org/10.1016/j.asoc.2014.09.025 1568-4946/© 2014 Elsevier B.V. All rights reserved.

Please cite this article in press as: R. Parameshwaran, et al., An integrated fuzzy MCDM based approach for robot selection considering objective and subjective criteria, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.09.025

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2 Table 1 MCDM techniques for robot selection. Techniques/tools used

Authors

Publication year

Selection criteria

AHP/fuzzy AHP

Liang and Wang [7] Goh [8] Rossetti and Selandari [9] Kapoor and Tak [10] Anand et al. [11] Athawale and Chakraborty [12]

1993 1997 2001 2005 2008 2011

I˙ c¸ et al. [4]

2013

Cost, repeatability, load capacity, velocity. Repeatability, cost, load capacity, velocity. Technical, economical, social, human, and environmental factors. Cost, velocity, repeatability, load capacity, stability, compliance, accuracy. Subjective and objective factors. Load capacity, repeatability, maximum tip speed, memory capacity and manipulator reach. Pay load, repeatability, vertical reach, robot mass, axis motion range, axis maximum speed.

Chu and Lin [13]

2003

Bhangale et al. [14]

2004

Kahraman et al. [15]

2007

Shih [16] Vahdani et al. [17]

2008 2011

Athawale and Chakraborty [12] Vahdani et al. [18] Rashid et al. [19]

2011 2013 2014

Chatterjee et al. [20]

2010

Athawale and Chakraborty [12] Devi [21]

2011 2011

Athawale et al. [22]

2012

Chatterjee et al. [20]

2010

Athawale and Chakraborty [12]

2011

PROMETHEE II

Athawale and Chakraborty [12]

2011

Load capacity, Repeatability, Maximum tip Speed, Memory capacity and Manipulator reach.

DEA

Braglia and Petroni [23]

1999

Talluri and Yoon [24] Wang and Chin [25] Athawale and Chakraborty [12] Mondal and Chakraborty [26]

2000 2009 2011 2013

Diameter, elevation, basic rotation, roll, pitch, yaw, cost, load capacity, repeatability and velocity. Cost, repeatability, load capacity, velocity. Purchasing cost, handling coefficient, load capacity, repeatability, velocity. Load capacity, repeatability, maximum tip speed, memory capacity, manipulator reach. Repeatability, load capacity, maximum tip speed, memory capacity, manipulator reach cost, handling coefficient, velocity.

TOPSIS/Fuzzy TOPSIS

VIKOR/Fuzzy VIKOR

ELECTRE II

Man–machine interface, purchase cost, programming flexibility, load capacity, vendor’s service contract, positioning accuracy. Load carrying capacity, repeatability, maximum tip speed, memory capacity, manipulator reach. Economical attributes: Investment costs – purchase cost, special tooling costs. Operating costs (OC) – maintenance cost, labor cost, training cost. Technical attributes: Repeatability, speed, memory capacity, precision or accuracy, programmability, number of axes, workload. Velocity, load capacity, cost, repeatability. Man–machine interface, purchase cost, programming flexibility, load capacity, vendor’s service contract, positioning accuracy. Load capacity, repeatability, maximum tip speed, memory capacity, manipulator reach. Purchase cost, load capacity, positioning accuracy. Man–machine interface, purchase cost, programming flexibility, load capacity, vendor’s service contract, positioning accuracy. Load capacity, repeatability, maximum tip speed, memory capacity, manipulator reach, velocity, cost, vendor’s service quality, programming flexibility. Load capacity, repeatability, maximum tip speed, memory capacity, manipulator reach. Man–machine interface, programming flexibility, vendor’s service contract, purchase cost, load capacity, positioning accuracy. Load capacity, repeatability, maximum tip speed, memory capacity, manipulator reach. Load capacity, repeatability, maximum tip speed, memory capacity, manipulator reach, velocity, cost, vendor’s service quality, programming flexibility. Load capacity, repeatability, maximum tip speed, memory capacity, manipulator reach.

Table 2 Integrated approaches for robot selection. Techniques/tools used

Authors

Publication year

Selection criteria

MCDM and computer aided robot selection OCRA, DEA, utility model and TOPSIS OCRA and TOPSIS

Agrawal et al. [27]

1991

Repeatability, cost, load capacity, velocity.

Parkan and Wu [28]

1999

Repeatability, cost, load capacity, velocity.

Bhangale et al. [14]

2004

QFD and AHP

Bhattacharya et al. [29]

2005

QFD and fuzzy linear regression Fuzzy subjective and objective integrated MADM method

Karsak [30]

2008

Load capacity, repeatability, maximum speed, type of drives, memory capacity, manipulator reach, degree of freedom. Drive system, geometrical dexterity, path measuring system, robot size, material of robot, weight of robot, initial operating cost, pay load, accuracy, life-expectancy, velocity of robot, programming flexibility, total cost. Load capacity, repeatability, vertical reach, horizontal reach, warranty period.

Rao et al. [3]

2011

Singh and Rao [31]

2011

Koulouriotis and Ketipi [32] Tao et al. [33]

2011

Graph theory and matrix approach (GTMA) and AHP Fuzzy digraph method and Fuzzy TOPSIS DEA, AHP, TOPSIS, AFS (axiomatic fuzzy set) theory

2012

Load capacity, repeatability, maximum tip speed, memory capacity, manipulator reach, velocity ratio, degree of freedom, purchase cost, repeatability error, man–machine interface, programming flexibility, service contract. Purchasing cost, handling coefficient, load capacity, velocity, repeatability. Man–machine interface, programming flexibility, repeatability error, purchase cost, velocity. Cost, load capacity, velocity, repeatability error.

Please cite this article in press as: R. Parameshwaran, et al., An integrated fuzzy MCDM based approach for robot selection considering objective and subjective criteria, Appl. Soft Comput. J. (2014), http://dx.doi.org/10.1016/j.asoc.2014.09.025

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Table 3 A general category of solutions for robot selection. Techniques/tools used

Authors

Publication year

Selection criteria

Computer aided robot selection procedure (CARSP) DSS

Offodile et al. [34]

1997

Workspace

Layek and Lars [35]

2000

Khouja et al. [36] Rao and Padmanabhan [37] Kumar and Garg [2]

2000 2006

Robot type (articulated, scara, etc.), degrees of freedom, pay load, horizontal reach, vertical reach, velocity, repeatability, power supply, program steps, memory size, control system, cost. Traveling time, rest time, repeatability, reach capacity, load capacity. Load capacity, repeatability, vertical reach, degree of freedom.

Kentli and Kar [38]

2011

Chakraborty [39]

2011

Load carrying capacity, repeatability, maximum tip speed, memory capacity, manipulator reach.

Karsak et al. [40]

2012

Cost, velocity, repeatability, load capacity

Vahdani et al. [41]

2014

Man–machine interface, load capacity, programming flexibility, positioning accuracy, vendor’s service contract, purchase cost.

Statistical approach Digraph and matrix methods Distance based approach Satisfaction function and distance measure approaches Multi-objective optimization on the basis of ratio analysis (MOORA) Fuzzy regression-based decision-making approach Interval-valued fuzzy multiple criteria complex proportional assessment (IVF-COPRAS)

64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

101

102 103

2010

selection criteria. From all the approaches, various criteria considered for selecting a robot for a particular application includes: load capacity, repeatability, horizontal and vertical reach, velocity or speed of travel, degree of freedom, positioning accuracy, maximum tip speed, memory capacity, manipulator reach, velocity ratio, purchase cost, repeatability error, warranty period, vendor’s service quality, programming flexibility, man–machine interface, stability, compliance, service contract, drive system, geometrical dexterity, path measuring system, robot size, material of robot, weight of robot, initial operating cost and life-expectancy. The literature review demonstrates that the majority of researchers have considered the selection of proper criteria as important, but they did not justify their selection or analyze their suitability [4]. Hence, a proper methodology to justify the selection of objective and subjective factors has to be incorporated in the decision making process. Fuzzy Delphi Method can be employed to overcome this limitation. The criteria which are defined in tangible terms are classified as quantitative or objective criteria e.g. cost, load capacity etc. and the criteria that have qualitative definitions i.e., they are unable to be defined in tangible terms are classified as qualitative or subjective criteria e.g. programming flexibility, stability, etc. The classification of criteria into objective and subjective for robot selection is done by only few authors [11,3,32,37]. Even though few researchers have considered this type of classification, weights given to objective and subjective factors are equal. But in real-life applications, the weights for quantitative and qualitative evaluation criteria will be different. Hence in the developed approach, this limitation is taken care off. Therefore, this paper aims to develop a new decision making method which takes care of suitable objective and subjective criteria selection and proper evaluation of the alternatives treating it as a MCDM problem. The proposed approach integrates Fuzzy Delphi Method, Fuzzy AHP, Fuzzy TOPSIS/Fuzzy VIKOR and Brown–Gibson model. In order to do so, the remainder of this paper is set out as follows. The proposed methodology is described in Section 2. In Section 3, selection of robot for educational purpose is used to illustrate the proposed method. Conclusions are presented in Section 4.

Velocity or speed of travel, repeatability, load capacity, degree of freedom, stability, compliance, accuracy. Cost, load capacity, speed, repeatability.

Identification of objective and subjective factors through literature review & Experts’ survey

Critical factors identification using Fuzzy Delphi method (FDM)

Weights calculation for objective and subjective factors using Fuzzy AHP method

Alternatives evaluation using Fuzzy modified TOPSIS / Fuzzy VIKOR method considering objective factors

Alternatives evaluation using Fuzzy modified TOPSIS / Fuzzy VIKOR method considering subjective factors

Robot selection index calculation using Brown-Gibson Model

Fig. 1. An integrated approach for robot selection.

critical factors identification using FDM, criteria weights calculation using FAHP Method, ranking of alternatives using Fuzzy Modified TOPSIS/Fuzzy VIKOR method and selection index calculation using Brown–Gibson model. The stages in the proposed approach are explained in the following sections. 2.1. Step I: listing the objective and subjective factors From the literature review and experts’ survey, the possible objective factors (OFs) and subjective factors (SFs) are listed. 2.2. Step II: critical factors identification using FDM FDM is applied to select the most critical OFs and SFs from the factors lists. The steps are as follows [42]:

104 105 106 107 108

109

110 111

112

113 114

2. Proposed methodology A systematic procedure is proposed in Fig. 1 which incorporates five steps namely listing the objective and subjective factors,

(i) Form a committee of experts and employ a questionnaire to ask experts for their most pessimistic (minimum) value and the most optimistic (maximum) value of the importance of each

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2.3. Step III: criteria weights calculation using FAHP Method

di

Membership µ

li

FAHP [43] is used to obtain the weights of the critical OFs and SFs obtained from the Step II. The procedure for FAHP is as follows. Let X = {x1 , x2 ,. . ., xn } be an object set, and U = {u1 , u2 ,. . ., un } be a goal set. According to the method of extent analysis, each object is taken and extent analysis for each goal is performed, respectively. Therefore, m extent analysis values for each object can be obtained ˜ 1,M ˜ 2gi , . . ., M ˜ j , where, all the M ˜ j (i = with the following signs: M

hi

1 Gray Zone

µ*

gi

0 lim hil

lil

si

liu

h im

P

h iu

Step 1: The value of fuzzy synthetic extent with respect to the ith object is defined as

OF (SF) in the possible factor set Sc (Se ) in a range from 1 to 10. A score for a OF (SF) is denoted as:

120

Ci = (lki , hik ),

i ∈ Sc

(1)

121

Ev = (lkv , hvk ),

v ∈ Se

(2)

122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138

gi

1, 2, . . ., n; j = 1, 2, . . ., m) are Triangular Fuzzy Numbers (TFNs). The steps of extent analysis can be given as,

m 

⎡ ⎤−1 m n   ˜ j ⊗⎣ ˜j ⎦ M M gi gi

j=1

i=1 j=1

Fig. 2. Gray zone.

119

gi

159 160 161 162 163 164 165 166 167

Cognition Value

gi

118

158

where lki (lkv ) is the pessimistic index of factor i(v) and hik (hvk ) is the optimistic index of factor i(v), rated by expert k. (ii) The triangular fuzzy number for the most pessimistic index and the most optimistic index for each factor i(v) is determined. The i (hi ) minimum value (lli (hil ) and llv (hvl )), the geometric mean (lm m

v (hv )) and the maximum value (li (hi ) and lv (hv )) of the and lm m u u u u

experts’ opinions on the most pessimistic index and most optii , li ), ·lv = (lv , lv , lv ) mistic index are obtained i.e., li = (lli , lm m u u l

and hi = (hil , him , hiu ), hv = (hvl , hvm , hvu ). (iii) The consensus of experts’ opinions is examined and the consensus significance value of each factor is calculated. The gray zone (Fig. 2) is the overlap section of li and hi and is used to examine the consensus of experts in each factor. The consensus significance value of the factor i, si , is calculated by the following rules: • If there is no overlap between li and hi (no gray zone exists), then consensus significance value of the factor is

S˜ i =

m

(6)

j

˜ perform the fuzzy addition operation of m M To obtain j=1 gi extent analysis values for a particular matrix such that

⎡ ⎤−1 m m m m     ˜j =⎣ M lj , mj , uj ⎦ gi j=1

j=1

j=1



(7)

˜j = M gi

i=1 j=1



 n n   li ,

i=1

169

170

171 172

173

j=1

n m ˜ j the fuzzy addition operation M ˜ j (j = M and to obtain i=1 j=1 gi gi 1, 2, . . ., m) values is performed such as m n  

168

mi ,

n 

i=1



ui

(8)

gi

u i=1 i

i=1 j=1

175

176

i=1

and then inverse of the above vector is computed such as

⎡ ⎤−1

m n   1 j ⎣ ⎦ ˜ = n M

174

,

1

n

i=1

mi

,

1

177



n

(9)

l i=1 i

As shown above, the value of the fuzzy synthetic extent with respect to the ith object is defined and S1 , S2 etc. are calculated.

178

179 180

i

139

140 141

l + him s = m 2 i

(3)

• If a gray zone exists and the gray zone interval value of gi (g i = lui − hil ) is less than the interval value of li and hi (di = him −

143

i ), i.e., gi ≤ di , the consensus significance value of the factor is lm calculated by (4) and (5).

144

F i (p) =

142



,

i∈S

146 147 148 149 150 151 152 153 154 155 156 157

s = {p/ max Fi (p), Fi (p)

1

(5) li

hi ,

si

is the area of the intersection of and and is where the cognition value (p) with the highest degree of membership (*) of the intersection of li and hi . • If a gray zone exists and gi > di , then a great discrepancy among the experts opinions arises. Above all steps need to be repeated until a convergence is obtained. The same procedure is carried out to calculate the consensus significance value of factor v, sv . (iv) Select factors from the factors list. Select factor i(v) if its consensus significance value is greater than or equal to threshold value TC (TE ) which is determined by the experts subjectively based on the mean of all si (sv ) i.e., select factor i(v) if si ≥ TC (sv ≥ TE ).

(10)

2

and can be equivalently expressed as follows:

˜ 2 ≥M ˜ 1 ) = (d) = V (M i∈S



˜ 2 ≥M ˜ 1 ) = supy≥x min  ˜ (x), min ˜ (y) V (M M M

(4)

p 145





{min[li (p), hi (p)]}dp

i

˜ 1 and M ˜ 2 are two triangular fuzzy numbers, the degree Step 2: As M ˜2 ≤M ˜ 1 is defined as of possibility of M

⎧ 1, ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

d (Ai ) = min V (S˜ i ≥S˜ K )

182

183

184

if m2 ≥m1

0,

if l2 ≥u2

l1 − u2 , (m2 − u2 ) − (m1 − l1 )

otherwise

(11)

where d is the ordinate of the highest intersection point D between ˜ 1 and M ˜ 2 , both values M˜ and M˜ as shown in Fig. 3. To compare M 1 2 ˜ 2 ≥M ˜ 1 ) and V (M ˜ 1 ≥M ˜ 2 ) are needed. of V (M 

181

(12)

The formulae required to compare two triangular fuzzy numbers are given in Step 2 and the degree of possibility is calculated in the Step 3. Step 3: The degree of possibility for a convex fuzzy number to be ˜ i can be defined by greater than k convex fuzzy numbers M

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186 187 188 189 190 191 192

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   m 2 ∗ where e =  c .

µ M~ (x)

j

1

~

0

X m2

l2

l1

d

u2

m1

v˜ ij = wj · r˜ij ,

195

˜ M ˜ 1, M ˜ 2 , . . ., M ˜ k ) = min V (M≥ ˜ M ˜ i ), V (M≥

where i = 1, 2, . . ., k. (13)

197

200

201

202

203

d (Ai )

Assume that = min V (S˜ i ≥S˜ K ) for k = 1, 2,. . ., k = / i. Then the weight vector is given by 







W = (d (A1 ), d (A2 ), . . ., d (An ))

T

(14)

j = 1, 2, . . ., n

230 231 232 233

234

1 (a + b + c) 3

(18)

Step 7: The positive ideal and negative ideal solutions are determined. The values for A* and A− are defined as: ∗

where Ai (i = 1, 2,. . ., n) are n elements.

i = 1, 2, . . ., m;

Step 6: The fuzzy numbers in the normalized decision matrix are defuzzified using centroid method. For the triangular fuzzy numbers, v˜ = (a, b, c), the defuzzified centroid value is:

v¯ =

198 199

Step 5: The fuzzy weighted normalized decision matrix, V˜ = [˜vij ]mxn . The fuzzy weighted normalized decision matrix is calculated by multiplying each column matrix by the weights obtained from FAHP.

u1

˜ 1 and M ˜ 2. Fig. 3. The intersection between M

196

229

ij

i=1

~ M1 )

V ( M2

5

A =

 = {(maxv˜ ij /j ∈ J), (minv˜ ij /j ∈ J ) i = 1, 2, . . ., m }

(v∗1 , v∗2 , . . ., v∗n )



235 236 237

238

239 240 241 242

i

i

(19)

Step 4: Via normalization, the normalized weight vectors are

243 244

T

204

W = (d(A1 ), d(A2 ), . . ., d(An ))

205

where W is a non-fuzzy number.

(15)

−

A

=

(v− , v− , . . ., v− n) 1 2

= {(minv˜ ij /j ∈ J),

245

i



(maxv˜ ij /j ∈ J  ) i = 1, 2, . . ., m }

(20)

246

i

206 207

208 209 210 211 212 213 214

215 216 217 218 219

2.4. Step IV: alternatives evaluation using Fuzzy modified TOPSIS/Fuzzy VIKOR method 2.4.1. Step IV (a): Fuzzy modified TOPSIS method Fuzzy TOPSIS method is used to rank the robots by considering OFs and SFs separately. In this paper, Fuzzy modified TOPSIS method proposed by Vahdani et al. [17] is used with two modifications: instead of incorporating both OF and SF into a single matrix, two different matrixes are prepared; the weight for each criterion is obtained from FAHP procedure. The steps are: Step 1: A group of k decision makers (DMs) is established. Step 2: The rating of m alternatives versus each of OF and SF criterion from each DM is obtained. Step 3: Let the ratings of the alternatives versus each OF criterion be x˜ ij .

222

The fuzzy ratings of all DMs are in the form of triangular fuzzy numbers x˜ ijk = (aijk , bijk , cijk ), k = 1, 2, . . ., K. The aggregated fuzzy rating can be obtained as:

223

x˜ ij = (aij , bij , cij ),

220 221

224

k = 1, 2, . . ., K

(16)

aij =

1 aijk , K

1 bijk , K K

bij =

k=1

J = {j = 1, 2, . . ., n|j associated with benefit attribute},

248

J  = {j = 1, 2, . . ., n|j associated with cost attribute},

249

The DM desires to have a maximum value among the alternatives for benefit criterion and a minimum value for cost criterion. Step 8: Construct ideal separation matrix (D*) and anti-ideal separation matrix (D− ) which are defined as:

⎡

1 cijk K

k=1

227

228

r˜ij =

aij bij cij , , ej∗ ej∗ ej∗

|¯vm1 − v¯ ∗1 |

⎡ −

[dij− ]

 i = 1, 2, . . ., m;

j = 1, 2, . . ., n

···

|¯v11 − v¯ − 1|

···

···

···

···

···

Si (D∗ , D− ) = ⎝

k  dij∗

(17)

dij−

250 251 252

253 254

⎤

|¯vmn − v¯ ∗n | |¯v1n − v¯ − n|

(21)

255

(22)

256

⎤

⎢ |¯v − v¯ − | · · · · · · |¯v − v¯ − | ⎥ 2n ⎢ 21 1 n ⎥ ⎥ =⎢ ⎢ ⎥ .. .. .. .. ⎣ ⎦ . . . .

i=1(A)

,

|¯v1n − v¯ ∗n |

|¯vmn − v¯ − n|

Step 9: The collective index (CI) is calculated using Eq. (23).

k=1

Step 4: The normalized decision matrix is computed. Vector normalization is applied to calculate r˜ ij .

···

⎢ |¯v − v¯ ∗ | · · · · · · |¯v − v¯ ∗ | ⎥ 2n ⎢ 21 1 n ⎥ ⎥ ⎥ .. .. .. .. ⎣ ⎦ . . . .

⎛ 226

···

|¯vm1 − v¯ − 1|

K

cij =

|¯v11 − v¯ ∗1 |

D∗ = [dij∗ ] = ⎢ ⎢

D =

where K

225

247

where

257

⎞1/k ⎠

+ Zij



where the first summation ( while (Zij ) refers to all

j

A)

∀i = 1, 2, . . ., m

(23)

258

refers to all j for which dij− > 0

259

for which dij− = 0. Moreover, Z ij can be

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6 Table 4 List of OFs and SFs.

261 262

263

Table 7 Triangular fuzzy numbers for FAHP comparisons.

Objective factors (OFs)

Subjective factors (SFs)

Linguistic terms

Triangular fuzzy numbers

Equipment cost Load capacity Repeatability Maximum tip speed Positioning accuracy Memory capacity Manipulator reach Degree of freedom Repeatability error Velocity ratio Velocity Vertical reach Warranty Delivery time Ambient temperature

Man–machine interface Programming flexibility Vendor’s service contract Supporting channel partner’s performance Simulation software Stability Compliance

Perfect Absolute Very good Fairly good Good Preferable Not bad Weak advantage Equal

(8, 9, 10) (7, 8, 9) (6, 7, 8) (5, 6, 7) (4, 5, 6) (3, 4, 5) (2, 3, 4) (1, 2, 3) (1, 1, 1)

and Ti . Minimum values of the CI indicate the better alternative. The same procedure is repeated for ranking the alternatives based on SFs.

calculated such that Zij = (maxj (dij∗  /dij− )) 0 and wj for dij− = 0.

Ti (D∗ , D− ) =

 k 1/m  dij∗

⎛

+ Qij

264

266

2.4.2. Step IV (b): Fuzzy VIKOR method The highest ranked alternative by TOPSIS is the best in terms of the ranking index, which does not mean that it is always closest to the ideal solution. Hence a new approach based on Fuzzy VIKOR method [44] is also used to obtain the rankings. This method focuses on ranking and selecting from a set of alternatives and determines compromise solutions for a problem with conflicting criteria, which can help the decision makers to reach a final decision. The steps are follows:

for which dij− >

⎞1/k

k  1 ⎠ +⎝ −

d

i=1(A) ij

i=1

265

1/(maxj wj )

∀i = 1, 2, . . ., m

(24)



where the second summation ( A ) refers to all j for which dij− > 0 while (Qij ) refers to all j for which dij− = 0 and Qij =

267

maxj wj (minj (dij− ))

268

index is calculated as follows:

269

CI = Si + Ti

for which dij−

>

0 and wj for dij−

Steps 1–6: The same steps as followed in Fuzzy TOPSIS method. Step 7: Determine the best fj∗ and the worst fj− values of all criterion ratings, j = 1, 2, . . ., n.

= 0. The collective (25)

272 273 274

275 276 277 278 279 280 281 282 283

284 285 286

fj∗ = maxxij ;

(26)

287

fj− = minxij .

(27)

288

i

270 271

Step 10: The preference order is ranked. The best satisfactory alternative can be determined according to preference rank order of Si

i

Table 5 FDM for OFs. S.No.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Objective factors (OFs)

Load capacity Equipment cost Repeatability Maximum tip speed Positioning accuracy Memory capacity Manipulator reach Degree of freedom Repeatability error Velocity ratio Velocity Vertical reach Warranty Delivery time Ambient temperature

Pessimistic value

Optimistic value

lli

i lm

lui

hil

him

hiu

5 4 4 6 4 5 2 2 1 2 4 5 1 4 1

6.38 5.96 5.27 6.78 6.50 6.45 4.69 4.23 1.73 3.55 4.61 6.29 4.09 5.88 1.94

8 8 8 8 8 7 7 6 3 7 6 7 8 7 3

7 7 7 7 7 6 5 5 2 5 5 6 5 6 2

7.78 7.78 7.78 7.96 8.45 7.89 7.30 7.17 2.83 6.90 6.19 8.05 7.51 7.44 3.91

9 9 9 9 10 10 10 9 4 10 8 10 10 9 6

a

di − gi

(sv )

0.41 0.82 1.51 0.18 0.95 0.44 0.61 1.94 0.10 1.35 0.58 0.76 0.42 0.56 0.97

7.35 7.28 7.21 7.45 7.5 6.75 6 5.55 2.4 5.75 5.47 6.73 6.23 6.6 2.73

Table 6 FDM for SF. S.No.

1 2 3 4 5 6 7

Subjective factors (SFs)

Man–machine interface Stability Simulation software Vendor service contract Programming flexibility Supporting channel partner performance Compliance

Pessimistic value

Optimistic value

lli

i lm

lui

hil

him

hiu

5 6 4 6 5 5 1

6.03 6.78 5.96 6.78 6.45 5.77 1.41

8 8 8 8 9 7 2

7 7 6 7 8 6 2

7.92 8.55 8.54 8.77 8.64 7.11 2.57

10 10 10 10 10 8 4

a

di − gi

(sv )

0.89 0.78 0.58 0.99 1.19 0.34 1.16

7.12 7.43 7.19 7.49 8.20 6.60 1.99

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Table 8 Pairwise comparison matrix for OFs.

O1 O2 O3 O4 O5 O6 O7 O8

O1

O2

O3

O4

O5

O6

O7

O8

Weight

(1, 1, 1) (2, 3, 4) (8, 9, 10) (1/6, 1/5, 1/4) (4, 5, 6) (1/7, 1/6, 1/5) (1/7, 1/6, 1/5) (1/7, 1/6, 1/5)

(1/4, 1/3, 1/2) (1, 1, 1) (2, 3, 4) (6, 7, 8) (1/5, 1/4, 1/3) (1/5, 1/4, 1/3) (1/9, 1/8, 1/7) (1/8, 1/7, 1/6)

(1/10, 1/9, 1/8) (1/4, 1/3, 1/2) (1, 1, 1) (1/5, 1/4, 1/3) (1/3, 1/2, 1/1) (1/10, 1/9, 1/8) (1/6, 1/5, 1/4) (1/9, 1/8, 1/7)

(4, 5, 6) (1/8, 1/7, 1/6) (3, 4, 5) (1, 1, 1) (5, 6, 7) (1/9, 1/8, 1/7) (4, 5, 6) (1/8, 1/7, 1/6)

(1/6, 1/5, 1/4) (3, 4, 5) (1, 2, 3) (1/7, 1/6, 1/5) (1, 1, 1) (1/9, 1/8, 1/7) (1/8, 1/7, 1/6) (1/10, 1/9, 1/8)

(5, 6, 7) (3, 4, 5) (8, 9, 10) (7, 8, 9) (7, 8, 9) (1, 1, 1) (3, 4, 5) (1/5, 1/4, 1/3)

(5, 6, 7) (7, 8, 9) (4, 5, 6) (1/6, 1/5, 1/4) (6, 7, 8) (1/5, 1/4, 1/3) (1, 1, 1) (1/5, 1/4, 1/3)

(5, 6, 7) (6, 7, 8) (7, 8, 9) (6, 7, 8) (8, 9, 10) (3, 4, 5) (3, 4, 5) (1, 1, 1)

0.09 0.16 0.37 0.06 0.31 0.00 0.00 0.00

Table 9 Pairwise comparison matrix for SFs.

S1 S2 S3 S4 S5 S6

S1

S2

S3

S4

S5

S6

Weight

(1, 1, 1) (7, 8, 9) (5, 6, 7) (4, 5, 6) (4, 5, 6) (6, 7, 8)

(1/9, 1/8, 1/7) (1, 1, 1) (1/3, 1/2, 1/1) (1/3, 1/2, 1/1) (1/5, 1/4, 1/3) (4, 5, 6)

(1/7, 1/6, 1/5) (1, 2, 3) (1, 1, 1) (1/5, 1/4, 1/3) (1/5, 1/4, 1/3) (4, 5, 6)

(1/6, 1/5, 1/4) (1, 2, 3) (3, 4, 5) (1, 1, 1) (1/5, 1/4, 1/3) (4, 5, 6)

(1/6, 1/5, 1/4) (3, 4, 5) (3, 4, 5) (3, 4, 5) (1, 1, 1) (4, 5, 6)

(1/8, 1/7, 1/6) (1/6, 1/5, 1/4) (1/6, 1/5, 1/4) (1/6, 1/5, 1/4) (1/6, 1/5, 1/4) (1, 1, 1)

0.00 0.24 0.16 0.00 0.00 0.59

where S ∗ = minSi , S − = maxSi , R∗ = minRi , R− = maxRi and v is

Table 10 Linguistic terms for rating alternatives.

289

290

i

Linguistic variables

Triangular fuzzy numbers

Very good (VG) Good (G) Medium good (MG) Fair (F) Medium poor (MP) Poor (P) Very poor (VP)

(9, 10, 10) (7, 9, 10) (5, 7, 9) (3, 5, 7) (1, 3, 5) (0, 1, 3) (0, 0, 1)

n 

wj (fj∗ − fij )/(fi∗ − fi− ),

(28)

295 296

297 298 299 300 301

302

Q (A(2) ) − Q (A(1) ) = DQ,

(31)

where A(2) is the alternative with second position in the ranking list by Q: DQ = 1/(J − 1).

Rj = maxwj (fj∗ − fij )/(fi∗ − fi− )

294

i

C1. Acceptable advantage:

j=1 291

i

Step 10: Rank the alternatives, sorting by the values S, R and Q in ascending order. Step 11: Propose as a compromise solution the alternative (A(1) ) which the best ranked by the measure Q(minimum) if the following two conditions are satisfied.

Step 8: Compute the values Si and Ri by the relations

Si =

i

introduced as a weight for the strategy of maximum group utility, whereas (1 − v) is the weight of the individual regret.

303

304 305

(29)

j

C2. Acceptable stability in decision making: 292

293

306

Step 9: Compute the values Qi by the relations Qj = v(Si − S ∗ )/(S − − S ∗ ) + (1 − v)(Ri − R∗ )/(R− − R∗ )

(30)

Table 11 Evaluation of alternatives with respect to OFs. Criterion

Alternatives

Decision makers

The alternative A(1) must also be the best ranked by S or/and R. This compromise solution is stable with in a decision making process, which could be the strategy of maximum group utility (when v > 0.5 is needed), or “by consensus” v ≈ 0.5, or “with veto” (v < 0.5). Here, v is the weight of decision making strategy of maximum group utility. If one of the conditions is not satisfied, then a set of compromise solution is proposed, which consists of

DM1

DM2

DM3

A1 A2 A3

MG F F

MG MG G

MG MP G

A1 A2 A3

MG MG MG

MG MG MG

MG MG MG

O3 (repeatability)

A1 A2 A3

MG G F

MG G F

MG G F

S2 (programming flexibility)

O4 (maximum TIP speed)

A1 A2 A3

MG MG G

MG F F

MG F F

O5 (positioning accuracy)

A1 A2 A3

MG G MG

MG G F

MG G F

O1 (equipment cost)

O2 (load capacity)

Table 12 Evaluation of alternatives with respect to SFs. Criterion

Alternatives

Decision makers DM1

DM2

DM3

A1 A2 A3

MG F MG

F F MG

F F G

S3 (vendors service contract)

A1 A2 A3

MG MG G

F F MG

F F MG

S6 (stability)

A1 A2 A3

MG MG VG

F F MG

F MG MG

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Table 13 Aggregated fuzzy ratings of the alternatives with respect to OFs. DM1

DM2

DM3

Aggregated ratings

O1

A1 A2 A3

5 3 3

7 5 5

9 7 7

5 5 7

7 7 9

9 9 10

5 1 7

7 3 9

9 5 10

5.00 3.00 5.67

7.00 5.00 7.67

9.00 7.00 9.00

O2

A1 A2 A3

5 5 5

7 7 7

9 9 9

5 5 5

7 7 7

9 9 9

5 5 5

7 7 7

9 9 9

5.00 5.00 5.00

7.00 7.00 7.00

9.00 9.00 9.00

O3

A1 A2 A3

5 7 3

7 9 5

9 10 7

5 7 3

7 9 5

9 10 7

5 7 3

7 9 5

9 10 7

5.00 7.00 3.00

7.00 9.00 5.00

9.00 10.00 7.00

O4

A1 A2 A3

5 5 7

7 7 9

9 9 10

5 3 3

7 5 5

9 7 7

5 3 3

7 5 5

9 7 7

5.00 3.67 4.33

7.00 5.67 6.33

9.00 7.67 8.00

O5

A1 A2 A3

5 7 5

7 9 7

9 10 9

5 7 3

7 9 5

9 10 7

5 7 3

7 9 5

9 10 7

5.00 7.00 3.67

7.00 9.00 5.67

9.00 10.00 7.67

Table 14 Aggregated fuzzy ratings of the alternatives with respect to SFs. DM1

315 316 317 318 319

DM2

DM3

A1 A2 A3

5 3 5

7 5 7

9 7 9

3 3 5

5 5 7

7 7 9

3 3 7

5 5 9

7 7 10

3.67 3.00 5.67

5.67 5.00 7.67

7.67 7.00 9.33

S3

A1 A2 A3

5 5 7

7 7 9

9 9 10

3 3 5

5 5 7

7 7 9

3 3 5

5 5 7

7 7 9

3.67 3.67 5.67

5.67 5.67 7.67

7.67 7.67 9.33

S6

A1 A2 A3

5 5 9

7 7 10

9 9 10

3 3 5

5 5 7

7 7 9

3 5 5

5 7 7

7 9 9

3.67 4.33 6.33

5.67 6.33 8.00

7.67 8.33 9.33

• Alternatives A(1) and A(2) if only the condition C2 is not satisfied, or • Alternatives A(1) , A(2) ,. . ., A(M) if the condition C1 is not satisfied; A(M) is determined by the relation Q(A(M) ) − Q(A(1) ) < DQ for maximum M (the positions of these alternatives are “in closeness”).

321

The same procedure is repeated for ranking the alternatives based on SFs.

322

2.5. Step V: selection index calculation

320

Aggregated ratings

S2

˜ i ) + (1 − ˛)(S˜ i ) ARSI = ˛(O

(32)

where 0 ≤ ˛ ≤ 1, ˛ = weight for objective factor; 1 − ˛ = weight for subjective factor. Step 3: Aggregated Robot Selection Index is defuzzified using centroid method to arrive at Robot Selection Index (RSI). The robot with highest RSI value is selected. 3. The case study – robot selection for educational purpose

323 324 325 326 327 328 329 330

Step 1: Triangular fuzzy numbers are assigned to the two different sets of ranking based on objective and subjective factors. Let them ˜ i and S˜ i respectively. be O Step 2: Each alternative’s fuzzy ranking is multiplied by its factor weight for objective and subjective factors respectively. Then, Brown and Gibson Model [45] is used to arrive at the Aggregated Robot Selection Index (ARSI). ARSI of ith alternative can be obtained through the following equation:

Due to vast growth in automation field, educational institutions have started to implement “Robotics” course in under graduate and post graduate studies. A robot for teaching purpose has to be selected in the department of Mechatronics Engineering. Quotations from three leading companies are called for and these three robots will be referred as A1 , A2 and A3 . From these three alternatives, the best Robot has to be selected. The following sections will explain the application of proposed methodology for the selection process.

Table 15 Normalized decision matrix for OF. Robots

O1

A1 A2 A3

0.34 0.21 0.39

O2 0.48 0.34 0.53

0.62 0.48 0.62

0.32 0.32 0.32

O3 0.45 0.45 0.45

0.58 0.58 0.58

0.33 0.46 0.20

O4 0.46 0.59 0.33

0.59 0.66 0.46

0.35 0.26 0.30

O5 0.49 0.40 0.44

0.63 0.54 0.56

0.32 0.45 0.24

0.45 0.58 0.37

0.58 0.65 0.50

Table 16 Normalized decision matrix for SF. Robots

S1

A1 A2 A3

0.26 0.21 0.41

S2 0.41 0.36 0.55

0.55 0.50 0.67

0.26 0.26 0.40

S3 0.40 0.40 0.54

0.54 0.54 0.65

0.25 0.30 0.43

0.39 0.43 0.55

0.52 0.57 0.64

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332 333

334 335 336

337

338 339 340 341 342 343 344 345 346

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Table 17 Defuzzified weighted normalized decision matrix for OF. Robots

O1

O2

O3

O4

O5

A1 A2 A3 R* (positive ideal solutions) R− (negative ideal solution)

0.045 0.032 0.048 0.032 0.048

0.072 0.072 0.072 0.072 0.072

0.173 0.214 0.123 0.214 0.123

0.029 0.024 0.026 0.029 0.024

0.141 0.175 0.115 0.175 0.115

Table 18 Defuzzified weighted normalized decision matrix for SF.

Table 22 Crisp values for decision matrix and weight of SF.

Robots

S1

S2

S3

A1 A2 A3 R* (positive ideal solution) R− (negative ideal solution)

0.097 0.086 0.130 0.130 0.086

0.067 0.067 0.089 0.089 0.067

0.228 0.255 0.318 0.318 0.228

Weight (from FAHP) A1 A2 A3

S1

S2

S3

0.24 5.67 5.00 7.56

0.17 5.67 5.67 7.56

0.59 5.67 6.33 7.89

3.3. Step III: weight calculation using Fuzzy AHP Method 347

348 349

350

351 352 353 354 355 356 357 358 359 360 361 362 363

364

3.1. Step I: listing the objective and subjective factors From the literature and the experts’ opinions the factors used for selecting robot are identified and listed in Table 4.

3.2. Step II: critical factors identification using FDM FDM is applied to select the most critical OFs and SFs from the factors lists. FDM is applied to OFs and SFs to find consensus significance value ((sv )a ). The results of FDM for OFs are shown in Table 5. With TE is set at 6.5, 8 OFs are shortlisted out of 15. The results of FDM for SFs are shown in Table 6. Here TE for SFs is set at 6.5. So, 6 SFs are shortlisted out of 7. From FDM, the selected OFs are: equipment cost (O1 ); load capacity (O2 ); repeatability (O3 ); maximum tip speed (O4 ); positioning accuracy (O5 ); memory capacity (O6 ); vertical reach (O7 ); delivery time (O8 ). Similarly the selected SFs are: man–machine interface (S1 ); programming flexibility (S2 ); vendor’s service contract (S3 ); supporting channel partner’s performance (S4 ); simulation software (S5 ); stability (S6 ).

Table 19 Ranking of robot under OF.

3.4. Step IV: alternatives evaluation using Fuzzy modified TOPSIS/Fuzzy VIKOR method

Si

Ti

CI

Final ranking

A1 A2 A3

12.0961 0 3.601483

9.194669 5.186998 8.201292

12.1053 5.186998 11.80277

3 1 2

Table 20 Ranking of robot under SF. Robots

Si

Ti

CI

Final ranking

A1 A2 A3

7.25 35.46 0

5.031163 4.263134 4.289867

12.28879 39.72737 4.289867

2 3 1

Table 21 Crisp values for decision matrix and weight of OFs. O1

O2

O3

O4

O5

0.09 7 5 7.44

0.16 7 7 7

0.37 7 8.67 5

0.06 7 5.67 6.22

0.31 7 8.67 6.33

A1

A2

A3

0.47 0.22 0.53 2

0.06 0.06 0.00 1

0.82 0.37 1.00 3

Table 24 S, R and Q values for each alternative with respect to SFs.

S R Q Ranking

366 367 368 369 370

372

3.4.1. Step IV (a): Fuzzy modified TOPSIS method The alternatives are evaluated using Fuzzy modified TOPSIS method. Table 10 shows the linguistic variables and their corresponding fuzzy numbers used in TOPSIS calculations. Three decision makers are asked to evaluate the alternatives with respect to OFs and SFs respectively using above mentioned linguistic terms. Tables 11 and 12 give the ratings of the robots with respect to selected OFs and SFs. Tables 13 and 14 show the aggregated fuzzy ratings of the alternatives by decision makers. The aggregated fuzzy rating is calculated using Eq. (16).

S R Q Ranking

365

371

Table 23 S, R and Q values for each alternative with respect to OFs.

Robots

Weight (from FAHP) A1 A2 A3

The weight of each selected factors are identified by using FAHP. Table 7 shows the linguistic variables and their corresponding fuzzy numbers used in FAHP calculations. Tables 8 and 9 denote the pairwise comparison matrix for OFs and SFs obtained from experts. The required weight of the factors used for robot selection process is calculated based on these tables.

A1

A2

A3

0.94 0.59 1.00 3

0.85 0.48 0.79 2

0.00 0.00 0.00 1

Table 25 Linguistic terms for compromised ranking. Rank

Linguistic variables

Triangular fuzzy numbers

1 2 3

High Medium Low

(0.5, 0.7, 0.9) (0.3, 0.5, 0.7) (0.1, 0.3, 0.5)

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373 374 375 376 377 378 379 380 381 382 383

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10 Table 26 RSI based on Fuzzy modified TOPSIS method. Alternatives

OF

A1 A2 A3

0.1 0.5 0.3

SF 0.3 0.7 0.5

0.5 0.9 0.7

0.5 0.7 0.3

0.7 0.9 0.5

0.3 0.1 0.5

ARSI 0.5 0.3 0.7

0.7 0.5 0.9

0.3 0.5 0.7

0.5 0.7 0.9

0.11 0.13 0.21

0.21 0.23 0.31

0.31 0.33 0.41

0.19 0.29 0.27

0.29 0.39 0.37

RSI

Rank

0.21 0.23 0.31

3 2 1

RSI

Rank

0.19 0.29 0.27

3 1 2

Table 27 RSI based on Fuzzy VIKOR.

384 385 386 387 388 389 390 391 392 393 394 395 396

397 398 399 400 401 402 403 404 405 406

Alternatives

OF

A1 A2 A3

0.3 0.5 0.1

SF 0.1 0.3 0.5

ARSI

The normalized decision matrix is calculated using Eq. (18). The normalized decision matrixes OFs and SFs are shown in Tables 15 and 16. The defuzzified weighted normalized decision matrix is calculated by multiplying each column of matrix by the weight, which is obtained by pair wise comparison of elements in the fuzzy AHP method. Consequently positive and negative ideal solutions are determined by Eqs. (19) and (20). Defuzzified weighted normalized decision matrix is shown in Tables 17 and 18. Ideal separation matrix (D*) and anti-ideal separation matrix (D− ) are constructed by using Eqs. (21) and (22). Finally, the values of Si , Ti and CI are computed by using Eqs. (23)–(25) and are shown in Tables 19 and 20.

3.4.2. Step IV (b): Fuzzy VIKOR method The aggregated fuzzy rating is calculated using Eq. (16). Then the aggregated fuzzy rating of alternatives is defuzzified into crisp value and listed in Tables 21 and 22. The weight of each criterion which is calculated from the fuzzy AHP method is also included in the crisp value decision matrix. The values of Si , Ri and Qi are calculated using Eqs. (28)–(30) for each alternatives separately for both OFs and SFs and are listed in Tables 23 and 24. The ranking using Fuzzy VIKOR method is also shown.

0.09 0.19 0.17

4. Conclusions Robots are preferred in many industrial applications to perform repetitious, difficult and hazardous tasks with precision. Hence the course “Robotics” is gaining momentum among the graduates. In this paper, a new methodology to select a robot for teaching robotic course is proposed. Initially fifteen quantitative and seven qualitative factors are considered for robot selection. FDM is used to select the potential criteria for further process based on the decision makers’ opinion. Fuzzy AHP method is then used to find out the weight of each criterion. Fuzzy modified TOPSIS/Fuzzy VIKOR method is then used to rank the alternatives based on OFs and SFs. The rankings obtained are converted into fuzzy numbers and are used to calculate the robot selection index using Brown–Gibson model. Weights for OF and SF helps to arrive at a better solution. Comparison between Fuzzy TOPSIS and Fuzzy VIKOR method results is studied and it is found that results from Fuzzy VIKOR method are closest to ideal solution. Through further refinements and extensive field testing, the proposed method can also be applied to various MCDM problems.

408

409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428

3.5. Step V: selection index calculation using Brown–Gibson model Since different set of rankings are obtained from Fuzzy TOPSIS and Fuzzy VIKOR based approach, the obtained ranks are converted into fuzzy numbers using Table 25. For education purpose robot, SFs get more weight than OFs. Therefore ˛ value is set as 0.4 i.e., (1 − ˛) = 0.6. Using Eq. (32), ARSI is calculated for each alternative and it is defuzzified using centroid method to arrive at RSI. The robot with highest RSI value is selected and results are presented in Tables 26 and 27. A3 is ranked as best alternative when Fuzzy modified TOPSIS method is used and A2 is ranked as best alternative when Fuzzy VIKOR based approach is used. The Fuzzy modified TOPSIS method introduces the ranking index including the distances from the ideal point and from the negative-ideal point. These distances are simply summed in without considering their relative importance. Hence, TOPSIS method has been proved to be less efficient. Moreover, the highest ranked alternative by Fuzzy VIKOR is the closest to the ideal solution. Fuzzy VIKOR method also proposes a compromise solution with an advantage rate. Hence, the result obtained by the Fuzzy VIKOR based method is taken as an optimal one. The robot A2 is selected as a best alternative among other three alternatives.

430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447

Acknowledgements

448

The authors would like to thank the referees who have contributed to enhancement of the technical contents of this paper. References

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