An Interval Type-2 Fuzzy Approach for Process Plan Selection

International Journal of Engineering Science Invention
ISSN (Online): 2319 – 6734, ISSN (Print): 2319 – 6726
www.ijesi.org ||Volume 4 Issue 6|| June 2015 || PP.14-22
www.ijesi.org 14 | Page
An Interval Type-2 Fuzzy Approach for Process Plan Selection
Anoop Kumar Sood1
,Tej Pratap Singh2
1
(Department Manufacturing Engineering, NIFFT, INDIA,)
2
(Department Manufacturing Engineering, NIFFT, INDIA,)
ABSTRACT: Process planning is a function in a manufacturing organization that systematically determines
the detailed methods such as the manufacturing processes and process parameters to be used to convert a part
from its initial design to the finished product. In a real manufacturing workplace the number of feasible
sequences for a part increases exponentially as the complexity of the product increases. The manufacturing of
several parts in a single facility sharing constrained resources and the existence of several alternative feasible
process plan for each part leads to careful selection of best process plan. This paper proposes the method of
process plan selection with the objective of minimizing the total processing time, total cost and the total number
of setup changes by using an interval type-2 fuzzy technique for order preference by similarity to Ideal
solutions. For this each process plan is evaluated and its likelihood closeness coefficient to shop floor
performance is calculated using interval type 2 fuzzy set theory.
KEYWORDS-Fuzzy, IT2TrF Numbers, Likelihood approach, Linguistic Variables, MCDA.
I. INTRODUCTION
Process Plan is set of instructions that are used to transform a component from initial raw material to
final finished product so that customer requirements are met.In doing so it translates design specifications into
manufacturing process details. Various attributes such as dimensions, geometry, and tolerances are continuously
transformed step by step to get final finished product in reasonable cost and limited time. Having various
options in alternative machines, alternative processes, and alternative setups to produce the same part, several
process plans can be generated for a single product. The process plan selection has become a critical problem
and such kind of problems is solved either by trial & error method or heuristic approaches [1]. In general
manufacturing cost, number of setups, processing steps, processing time and flow rate of parts are main
criteria’s for designing or selecting a good process plan [2,3]. Considering the intrinsic differences in
unevenness of raw material, machining parameters, and other manufacturing activities and information, the
process planning can be vague and contradictory in nature.In process planning problem objectives are
conflicting and information is imprecise and ambiguous. Considering this ambiguity and vagueness in process
plan selection problem fuzzy based approaches can be considered as natural solution procedure for process plan
selection. Fuzzy based approaches apply fuzzy logic to enumerate the role of each process plan to the shop floor
performance in terms of fuzzy membership function [4].
Selection of most optimum process plan on the basis of cost, machining time, machine setups etc. makes the
problem in category of multi criteria decision analysis problem. The technique for order preference by similarity
to ideal solutions (TOPSIS), introduced by Hwang and Yoon, is an extensively used method for handling
multiple criteria decision analysis (MCDA) problems [5]. In TOPSIS each criteria is given a performance rating
against each alternative and selection of best alternative depends upon the relative importance of criteria’s. In
most practical cases it is often difficult for decision makers to assign precise performance values to an
alternative with respect to a criteria or an accurate value of relative importance among criteria under
consideration. The advantage of using a fuzzy approach in the TOPSIS methodology is to assign the linguistic
ratings using fuzzy numbers instead of precise numbers [6]. The traditional fuzzy sets are represented by its
membership functions that are chosen for a specified criteria. But it is often difficult to quantify membership
function value as a number in interval [0,1]. Therefore, it is more suitable to represent this degree of certainty by
an interval. In this regard type 2 fuzzy sets are the extension of ordinary fuzzy set concept, in which the
membership function falls into an interval consisting of lower and upper limit of degree of membership
[7].Interval type-2 trapezoidal fuzzy (IT2TrF) numbers provide more reasonable and computationally feasible
method for handling complicated interval type-2 fuzzy data [7].Therefore, to formulateimprecisions and
uncertainties, this paper attempts to advance a TOPSIS based on IT2TrF numbers for quantifying the ambiguous
nature ofprocess plan selection problem. Positive ideal and negative ideal solutions are represented using
IT2TrF number and likelihood based comparison approach is adopted to rate different alternatives.Next section
introduces the basic definitions and notations of the IT2TrF numbers and linguistic variables with likelihood

approach. Section 3 shows the proposed algorithm for likelihood closeness coefficients calculation and rank
allocation. Section 4 contains process planning problem and then, the proposed method is illustrated with an
example. Finally, some conclusions are pointed out in the end of this paper.
II. INTERVAL TYPE-2 FUZZY SETS AND IT2TrF NUMBERS
Selected relevant definitions and properties are briefed here to explain the concepts of interval type-2
fuzzy sets and IT2TrF numbers used throughout this paper [8, 9, 10].
Definition 1.Let Int([0, 1]) be the set of all closed subintervals of [0, 1]. A mapping A: X→Int([0, 1]) is
known as an interval type-2 fuzzy set in X; where X is an ordinary finite nonempty set.
Definition 2.For type-2 fuzzy set A; lower fuzzy set is ]1,0[: 
XA and upper fuzzy set is ]1,0[
XA
.The value ]1,0[)](),([)(  
xAxAxA represent the degree of membership of Xx to A.
Definition 3.For IT2TrF the lower and upper membership functions )(xA
and )(xA
respectively are defined
as follows:











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
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
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

















;0
,
)(
,
,
)(
)(
43
34
4
32
21
12
1
otherwise
axaif
aa
xah
axaifh
axaif
aa
axh
xA
A
A
A
(1)

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
















.0
,
)(
,
,
)(
)(
43
34
4
32
21
12
1
otherwise
axaif
aa
xah
axaifh
axaif
aa
axh
xA
A
A
A
(2)
Where 
3214321 ,,,,,, aaaaaaa and 
4a are real and 
 114321 , aaaaaa and 
 44 aa . 
Ah and 
Ah
represents the heights of 
A and 
A respectively such as 10  
AA hh Then A is an IT2TrF number in X and
is expressed as follows:
)];,,,(),;,,,[(],[ 43214321

 AA haaaahaaaaAAA (3)
Definition 4.Let ],[ 
 AAA and ],[ 
 BBB be any two IT2TrF in X. Let  be a positive integer. Assume that
at least one of 
 1414 ,, bbaahh BA and 
  ba holds and at least one of 
 1414 ,, bbaahh BA and

  ba Where .4,3,2,1
The lower ( 
LI ) and upper ( 
LI ) likelihoodof an IT2TrF binary relation BA is defined as follows


























 
 







0,0,
2)()(
)0,max(2)()0,max(
max1max)( 4
1
1414
14
4
1




AB
AB
hhbbaaab
hhabab
BALI (4)
























 
 







0,0,
2)()(
)0,max(2)()0,max(
max1max)( 4
1
1414
14
4
1




AB
AB
hhbbaaab
hhabab
BALI (5)
The likelihood )( BAL  of an IT2TrF binary relation BA is given by the following:
2
)()(
)(
BALIBALI
BALI



(6)

This paper determines the lower likelihood of )( BAL  via the relation 
 BA because the minimal
possibility of the event BA  generally occurs in the comparison of 
A and 
B . Additionally, this paper
determines the upper likelihood of )( BAL  via the relation 
 BA because the maximal possibility of the
event BA . Generally occurs in the comparison of 
A & 
B . The proposed likelihood ),( BAL 
)( BAL 
and
)( BAL  in Definition 4 possess the following properties.
Property 1.The upper and lower likelihoods ),( BAL 
)( BAL 
respectively, of an IT2TrF binary relation
BA satisfy the following properties
).0,max(21)(0)()5(
;1)(0)()4(
;1)()()3(
;1)(0)2(
;1)(0)1(
41
14










BA
BA
hhabifABLandBAL
hhandbaifABLandBAL
ABLBAL
BAL
BAL
Property 2.The likelihood of )( BAL  an IT2TrF binary relation BA satisfies the following properties:
.5.0)()4(
);()(5.0)()()3(
;1)()()2(
;1)(0)1(




AAL
ABLBALifABLBAL
ABLBAL
BAL
III. PROPOSED ALGORITHM
This section develops the interval type 2 fuzzy TOPSIS method based on IT2TrF numbers.
Step 1: Formulate an MCDA problem. Specify the alternative set },...,{ 21 mzzzZ  and the criterion set
},...,{ 21 ncccC  which is divided into CI (benefit criteria) andCII(cost criteria).
Step 2: Select the appropriate linguistic variable giving a rating of selected alternative (zi) with respect to the
given criteria (cj) with appropriate IT2TrF number ijA define by (3).
Step 3: Define the relative importance or criteria weight jW in terms of IT2TrF number expressed as follows:
 
],[ jjj WWW )];,,,(),;,,,[( 43214321

jj WjjjjWjjjj hwwwwhwwww (7)
Step 4: Determine the weighted evaluative rating using (8)
     
  
  















jj
jj
ijij
AWjjjjjjjj
AWjjjjjjjj
AijijijijAijijijijijij
ijjij
hhawawawaw
hhawawawaw
haaaahaaaaAA
AWA
,min;,,,
,,min;,,,
;,,,,;,,,,
44332211
44332211
43214321
(8)
Step 5: Apply (9) and (10) to derive the weighted evaluative rating jA for the approximate positive-ideal
solution z with respect to criterion .Ccj  in (11)


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
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



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
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










































Ccifhaaaa
Ccifhaaaa
A
jA
m
i
ij
m
i
ij
m
i
ij
m
i
ij
m
i
jA
m
i
ij
m
i
ij
m
i
ij
m
i
ij
m
i
j
ij
ij
1
4
1
3
1
2
1
1
1
1
4
1
3
1
2
1
1
1
;,,,
,;,,,
 (9)

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
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

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
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


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



























Ccifhaaaa
Ccifhaaaa
A
jA
m
i
ij
m
i
ij
m
i
ij
m
i
ij
m
i
jA
m
i
ij
m
i
ij
m
i
ij
m
i
ij
m
i
j
ij
ij
1
4
1
3
1
2
1
1
1
1
4
1
3
1
2
1
1
1
;,,,
,;,,,
 (10)
)];,,,(),;,,,[(],[ 43214321


jj AjjjjAjjjjjjj haaaahaaaaAAA

 (11)
Step 6: Apply (12) and (13)to derive the weighted evaluative rating jA for the approximate negative-ideal
solution z with respect to criteria Ccj  in (14).










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


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

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













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

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
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

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










































Ccifhaaaa
Ccifhaaaa
A
jA
m
i
ij
m
i
ij
m
i
ij
m
i
ij
m
i
jA
m
i
ij
m
i
ij
m
i
ij
m
i
ij
m
i
j
ij
ij
1
4
1
3
1
2
1
1
1
1
4
1
3
1
2
1
1
1
;,,,
,;,,,
 (12)

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
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
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
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


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

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

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










































Ccifhaaaa
Ccifhaaaa
A
jA
m
i
ij
m
i
ij
m
i
ij
m
i
ij
m
i
jA
m
i
ij
m
i
ij
m
i
ij
m
i
ij
m
i
j
ij
ij
1
4
1
3
1
2
1
1
1
1
4
1
3
1
2
1
1
1
;,,,
,;,,,
 (13)
)];,,,(),;,,,[(],[ 43214321


jj AjjjjAjjjjjjj haaaahaaaaAAA

 (14)
Step 7: Determine the likelihood-based comparison indices )(),( jijjij AALIAALI   
and )( jij AALI  of
ijA relative to jA by using (15), (16) and (17) respectively for each Zzi  with respect to Ccj  .


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
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
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
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













 
 









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 )( jij AALI 
2
)()( jijjij AALIAALI   
(17)
Step 8: Determine the likelihood-based comparison indices )(),( jijjij AALIAALI   
and )( jij AALI  of ijA
relative to jA by using (18), (19) and (20) respectively for each Zzi  with respect to Ccj  .
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2
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Step 9: Derive the likelihood-based closeness coefficient iLC using (21)
  
 

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n
j
n
j
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n
j
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AALI
LC
1 1
1
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)(


(21)
For each alternative Zzi  with respect to Ccj  .
Step 10: Rank the m alternatives in accordance with the iLC values. The alternative with the largest iLC value is
the best choice.

IV. PROBLEM FORMULATION
Three sample parts from namely Job 1 from [11, 12], Job 2 from [13], and Job 3 from[14] are considered for
process plan selection problem. There are seven feasible process plan for Job 1, five feasible process plan for Job 2 and four
feasible process plan for Job 3considering criteria such as total cost, setup changes, machine changes and tool changes. Time
consumed in machining a job at various machines and tool combination is calculated from the data given in [15].Total cost
involved is the sum of the machine usage cost,tool usage cost,setup cost,machine change cost, tool change cost.Total time to
complete one process plan involves working time of machine, material handling time and setup time.The set of all
alternative process plans is denoted by Z {z1,z2,z3,z4,z5,z6,z7,z8,z9,z10,z11,z12,z13,z14,z15,z16}and five criteria’s cost(c1), setup
change(c2),tool Change(c3),machine change(c4) and Time (c5) and are given in Table I. All five criteria denote minimize
type.
Table I:Process Plans & their criteria for all the jobs
Job no.
Process
Plans
Cost
Setup
Change
Tool
change
Machine
Change
Time
1
z1 2537 10 9 2 473
z2 2535 10 9 2 483
z3 2527 10 10 2 587
z4 2567 10 9 2 625
z5 2720 8 16 1 454
z6 3215 7 15 1 543
z7 3205 6 15 0 516
2
z8 1739 3 8 1 644
z9 2664 11 13 0 543
z10 3799 0 10 8 550
z11 5014 5 0 12 677
z12 1784 3 9 1 649
3
z13 745 1 5 0 299
z14 1198 1 5 0 292
z15 833 2 5 0 319
z16 1308 2 3 2 378
V. RESULTS
In this paper five point linguistic rating scales is takento establish the evaluative ratings of the alternatives with
respect to criteria’s. These linguistic variables expressed in IT2TrF numbers and are given in Table II.
Table II: Linguistic terms and their corresponding interval Trapezoidal type-2 fuzzy sets
Linguistic terms Symbol Interval type-2 fuzzy sets
Very low (VL) VL ((0,0.1,0.2,0.3;1),(0,0.13,0.17,0.27;0.8))
Low (L) L ((0.1,0.2,0.3,0.4;1),(0.13,0.23,0.27,0.37;0.8))
Medium (M) M ((0.3,0.4,0.5,0.6;1),(0.33,0.43,0.47,0.57;0.8))
High (H) H ((0.5,0.6,0.7,0.8;1),(0.53,0.63,0.67,0.77;0.8))
Very high (VH) VH ((0.7,0.8,0.9,1;1),(0.73,0.83,0.87,0.97;0.8))
The relative weightage ratings of criteria in terms of IT2TrF number are presented in Table III.
Table III:Weightage Ratings for various Criteria
Criteria Weightage Ratings
Cost (c1) ((0.1,0.2,0.3,0.4;1),(0.13,0.23,0.27,0.37;0.8))
Setup change (c2) ((0.3,0.4,0.5,0.6;1),(0.33,0.43,0.47,0.57;0.8))
Tool change (c3) ((0.3,0.4,0.5,0.6;1),(0.33,0.43,0.47,0.57;0.8))
Machine change (c4) ((0.3,0.4,0.5,0.6;1),(0.33,0.43,0.47,0.57;0.8))
Time (c5) ((0.1,0.2,0.3,0.4;1),(0.13,0.23,0.27,0.37;0.8))
Linguistic performance rating of each alternative with respect to criteria is given in Table IV.

Table IV: Linguistic performance rating of alternatives and their relative ranking
Criteria
Job Process Plan c1 c2 c3 c4 c5 LC Rank
Job 1
z1 VL H VL M VL 0.5293135 1
z2 VL H VL M L 0.5188261 2
z3 VL H L M M 0.4872964 5
z4 L H VL M N 0.4756051 6
z5 M M H L VL 0.4998666 4
z6 VH L H L M 0.4720732 7
z7 VH L H VL L 0.5054797 3
Job 2
z8 VL L M VL M 0.5756089 1
z9 M VH VH VL L 0.4932366 4
z10 H VL H M M 0.5093507 3
z11 VH L VL VH VH 0.4712634 5
z12 VL L H VL H 0.5481439 2
Job 3
z13 VL M H VL M 0.5505214 1
z14 H M H VL L 0.5299625 2
z15 L H H VL M 0.5101461 3
z16 VH H L H VH 0.4293073 4
The IT2TrF numbers are utilized to perform the calculations for each alternative process plan
according to steps given in the algorithm.Table V shows the calculated values of positive and negative
likelihood based comparison indices of each criteria within a process plan.Likelihood based closeness
coefficients of each process plan can be obtained by using equation (21). Rank ordering for most optimum
process plan is done on the basis of Likelihood-based closeness coefficients obtained. For rank ordering, value
of likelihood closeness coefficient (LC) near to one is considered as best in each job. That is for job 1 process
plan (alternative) z1is having LC value maximum hence selected, similarly for job 2 process plan z8 is selected
and for job 3 process plan z13 is selected. LC values and their relative rankings are presented in Table IV.
Table V: Positive and negative Likelihood Values for various criteria in a process plan
Process
Plan
Criteria )( jAijALI 

)( jAijALI 

)( jAijALI  )( jAijALI 

)( jAijALI 

)( jAijALI 
z1
c1 0.165361 0.834639 0.5 0.709552 0.980762 0.845157
c2 0.704436 0.96283 0.833633 0.306128 0.693872 0.5
c3 0.213317 0.786683 0.5 0.764969 0.99785 0.881409
c4 0.641912 0.954167 0.79804 0.280382 0.719618 0.5
c5 0.165361 0.834639 0.5 0.628908 0.961443 0.795175
z2
c1 0.165361 0.834639 0.5 0.709552 0.980762 0.845157
c2 0.704436 0.96283 0.833633 0.306128 0.693872 0.5
c3 0.213317 0.786683 0.5 0.764969 0.99785 0.881409
c4 0.641912 0.954167 0.79804 0.280382 0.719618 0.5
c5 0.300862 0.88388 0.592371 0.571316 0.928206 0.749761
z3
c1 0.165361 0.834639 0.5 0.709552 0.980762 0.845157
c2 0.704436 0.96283 0.833633 0.306128 0.693872 0.5
c3 0.408219 0.872819 0.640519 0.704436 0.96283 0.833633
c4 0.641912 0.954167 0.79804 0.280382 0.719618 0.5
c5 0.506924 0.932412 0.719668 0.43208 0.847978 0.640029
z4
c1 0.300862 0.88388 0.592371 0.66626 0.954502 0.810381
c2 0.704436 0.96283 0.833633 0.306128 0.693872 0.5
c3 0.213317 0.786683 0.5 0.764969 0.99785 0.881409
c4 0.641912 0.954167 0.79804 0.280382 0.719618 0.5
c5 0.628908 0.961443 0.795175 0.280382 0.719618 0.5
z5
c1 0.506924 0.932412 0.719668 0.565172 0.893128 0.72915
c2 0.544355 0.900772 0.722563 0.540594 0.868139 0.704366
c3 0.764969 0.99785 0.881409 0.306128 0.715986 0.511057
c4 0.408219 0.872819 0.640519 0.544355 0.900772 0.722563
c5 0.165361 0.834639 0.5 0.628908 0.961443 0.795175

z6
c1 0.709552 0.980762 0.845157 0.306128 0.693872 0.5
c2 0.246752 0.753248 0.5 0.704436 0.96283 0.833633
c3 0.764969 0.99785 0.881409 0.306128 0.715986 0.511057
c4 0.408219 0.872819 0.640519 0.544355 0.900772 0.722563
c5 0.506924 0.932412 0.719668 0.43208 0.847978 0.640029
z7
c1 0.709552 0.980762 0.845157 0.306128 0.693872 0.5
c2 0.246752 0.753248 0.5 0.704436 0.96283 0.833633
c3 0.764969 0.99785 0.881409 0.306128 0.715986 0.511057
c4 0.213317 0.786683 0.5 0.641912 0.954167 0.79804
c5 0.300862 0.88388 0.592371 0.571316 0.928206 0.749761
z8
c1 0.165361 0.834639 0.5 0.709552 0.980762 0.845157
c2 0.408219 0.872819 0.640519 0.798597 0.999563 0.89908
c3 0.641912 0.954167 0.79804 0.691306 0.934208 0.812757
c4 0.213317 0.786683 0.5 0.840918 1 0.920459
c5 0.423908 0.887521 0.655715 0.545599 0.893128 0.719363
z9
c1 0.506924 0.932412 0.719668 0.565172 0.893128 0.72915
c2 0.840918 1 0.920459 0.326471 0.673529 0.5
c3 0.840918 1 0.920459 0.326471 0.698092 0.512281
c4 0.213317 0.786683 0.5 0.840918 1 0.920459
c5 0.200959 0.799041 0.5 0.652807 0.954502 0.803654
z10
c1 0.628908 0.961443 0.795175 0.438667 0.815758 0.627213
c2 0.213317 0.786683 0.5 0.840918 1 0.920459
c3 0.764969 0.99785 0.881409 0.53742 0.840421 0.688921
c4 0.641912 0.954167 0.79804 0.691306 0.934208 0.812757
c5 0.423908 0.887521 0.655715 0.545599 0.893128 0.719363
z11
c1 0.709552 0.980762 0.845157 0.306128 0.693872 0.5
c2 0.408219 0.872819 0.640519 0.798597 0.999563 0.89908
c3 0.213317 0.786683 0.5 0.840918 1 0.920459
c4 0.840918 1 0.920459 0.326471 0.673529 0.5
c5 0.66626 0.954502 0.810381 0.292119 0.693872 0.492995
z12
c1 0.165361 0.834639 0.5 0.709552 0.980762 0.845157
c2 0.408219 0.872819 0.640519 0.798597 0.999563 0.89908
c3 0.764969 0.99785 0.881409 0.53742 0.840421 0.688921
c4 0.213317 0.786683 0.5 0.840918 1 0.920459
c5 0.571316 0.928206 0.749761 0.414743 0.815758 0.615251
z13
c1 0.165361 0.834639 0.5 0.709552 0.980762 0.845157
c2 0.280382 0.719618 0.5 0.540594 0.868139 0.704366
c3 0.704436 0.96283 0.833633 0.306128 0.715986 0.511057
c4 0.213317 0.786683 0.5 0.764969 0.99785 0.881409
c5 0.423908 0.887521 0.655715 0.545599 0.893128 0.719363
z14
c1 0.628908 0.961443 0.795175 0.438667 0.815758 0.627213
c2 0.280382 0.719618 0.5 0.540594 0.868139 0.704366
c3 0.704436 0.96283 0.833633 0.306128 0.715986 0.511057
c4 0.213317 0.786683 0.5 0.764969 0.99785 0.881409
c5 0.200959 0.799041 0.5 0.652807 0.954502 0.803654
z15
c1 0.300862 0.88388 0.592371 0.66626 0.954502 0.810381
c2 0.540594 0.868139 0.704366 0.306128 0.693872 0.5
c3 0.704436 0.96283 0.833633 0.306128 0.715986 0.511057
c4 0.213317 0.786683 0.5 0.764969 0.99785 0.881409
c5 0.423908 0.887521 0.655715 0.545599 0.893128 0.719363
z16
c1 0.709552 0.980762 0.845157 0.306128 0.693872 0.5
c2 0.540594 0.868139 0.704366 0.306128 0.693872 0.5
c3 0.246752 0.753248 0.5 0.680036 0.96283 0.821433
c4 0.764969 0.99785 0.881409 0.306128 0.693872 0.5
c5 0.66626 0.954502 0.810381 0.292119 0.693872 0.492995

VI. CONCLUSION
This paper provides the interval type-2 fuzzy TOPSIS algorithmic procedure for determining the
priority ranking orders of alternative process plans for various jobs. The complexity in process plan selection
problem can be solved with IT2TrF numbers, which reduces ambiguity & vagueness. The results establish that
the proposed interval type-2 fuzzy TOPSIS method is effective and valid for addressing the process plan
problems in fuzzy environment. Although the proposed method presented in this paper is illustrated by a process
plan selection problem, however, it can also be applied to problems such as information project selection,
material selection and many other areas of multiple decision problems.
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An Interval Type-2 Fuzzy Approach for Process Plan Selection

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