INTRODUCTION
Cucumis melo L. is a tropical and sub-tropical fruit
that easily decays and rots because of its high-water
content (70–95%). For the fruit to maintain its quality
and freshness, it has to be handled properly during and
after harvesting. A good quality fruit should be fresh,
with a smooth, undamaged, and flawless skin. Compared
to other cucumbers (Cucumis), Cucumis melo has a
greener colour, more crunchy texture, higher water
content, and sweeter taste. In addition, Cucumis melo
can be harvested at an earlier stage.
Packaging is extremely important in post-harvest
handling. It creates proper condition for the fruit to
maintain its quality during the desired period. Packaging
is a container or wrapper that can help to prevent or
reduce damage to the packaged/wrapped object. The
main functions of packaging are to keep food products
from contamination, to protect them from physical
damage, and to inhibit their quality degradation.
In the Province of Aceh (Indonesia), Cucumis melo
is usually packaged in traditional manner by using
banana stem, because banana leaves are cheap, easy to
find, and eco-friendly. The fruit is placed in the middle
part of banana stem, which are then folded into two parts
(Fig. 1). Banana stem are able to protect the fruit from
shocks and damage during transportation from producer
to consumer. When ripe, the epidermis of Cucumis melo
cracks, and banana stem help keep its shape and texture.
Usually, Cucumis melo is protected with a single layer of
banana stem.
According to Lukman [1], banana stem is part of
banana pseudo stem. Its structure is very different from
that of woody plants, because it is an apparent trunk
formed by tightly packed, over-leaping stem. The fibre
of banana stem are strong and waterproof to both fresh
and salt water. The packaging of Cucumis melo with a
various amount of banana stem is necessary to preserve
its wholeness and texture, because this fruit is easily
broken when ripe. The storage temperature varies from
room temperature to cold temperature, which is also
expected to prolong the shelf life of Cucumis melo.
A quick method to find out consumer acceptance
towards the food product is to perform a sensory
assessment by collecting respondents’ opinions on the
product. This multi-criteria assessment method was
completed with a weighting assessment approach,
Research Article DOI: http://doi.org/10.21603/2308-4057-2019-2-339-347
Open Access Available online at http:jfrm.ru
A multi-criteria sensory assessment of Cucumis melo (L.) using
fuzzy-Eckenrode and fuzzy-TOPSIS methods
Rahmat Fadhil* , Raida Agustina
Universitas Syiah Kuala, Banda Aceh, Indonesia
* e-mail: rahmat.fadhil@unsyiah.ac.id
Received May 26, 2019; Accepted in revised form June 17, 2019; Published October 21, 2019
Abstract: The paper introduces a multi-criteria assessment system that can be used for sensory analysis by fuzzy-Eckenrode
and fuzzy-TOPSIS methods. Respondents evaluated the sensory characteristics of Cucumis melo (L.), which included aroma,
colour, taste, texture, and overall acceptance, after six days of storage. The product was stored under three different temperature
conditions: 10°C (B1), 14°C (B2), and room temperature (27–30°C) (B3). The product was also stored at three types of packaging:
unpackaged stem (A1), packaged fruit with one layer of banana stem (A2), and packaged fruit with two layers of banana stem
(A3). The best result was demonstrated by the Cucumis melo that was stored at 14°C and packaged in a two-layered banana stem
(A3B2). Both fuzzy-Eckenrode and fuzzy-TOPSIS method provided an easy, fast, and unambiguous calculation of multi-criteria
sensory assessment.
Keywords: Banana stem, hedonic scale, Cucumis melo (L.), sensory assessment, TOPSIS, Eckenrode
Please cite this article in press as: Fadhil R, Agustina R. A multi-criteria sensory assessment of Cucumis melo (L.) using
fuzzy-Eckenrode and fuzzy-TOPSIS methods. Foods and Raw Materials. 2019;7(2):339–347. DOI: http://doi.org/10.21603/2308-
4057-2019-2-339-347.
Copyright © 2019, Fadhil et al. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International
License (http://creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix,
transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license.
Foods and Raw Materials, 2019, vol. 7, no. 2
E-ISSN 2310-9599
ISSN 2308-4057
340
Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347
which is usually used in decision making. Therefore,
this article introduces a multi-criteria assessment
system that performs a sensory analysis by using fuzzy-
Eckenrode and the fuzzy-TOPSIS (Technique for Order
Performance by Similarity to Ideal Solution) methods.
According to the system, the respondents evaluated
each product and rated its level of acceptance according
to a multi-criteria sensory assessment, which included
aroma, colour, taste, texture, and overall acceptance.
Fuzzy logic. Fuzzy logic is a development of the set
theory, where each member has a degree of membership
that ranges in value between 0 and 1. It means that fuzzy
sets can represent interpretation of each value according
to the opinion, or decision, and its probability. Rating 0
represents ‘wrong’, rating 1 represents ‘right’, and there are
still other numbers between the ‘right’ and ‘wrong’ [2, 3].
In fuzzy sets, there are two attributes. The first one
is linguistic attribute: it is a naming of a group which
represents a certain situation or condition by using
a natural language such as ‘cold’, ‘cool’, ‘normal’, or
‘warm’. The second attribute is numeric: it is a value
(number) which shows a measure of a variable, such as
10, 30, 50, etc. [4]. Membership function is a curve that
defines how each point in the input room is mapped into
the membership value (degree of membership between 0
and 1). If U states universal sets and A is fuzzy function
sets in U, so A can be stated as sorted pair as following [2]:
{(x (x)) x U} A Α = ,μ ∈ (1)
where (x) A μ is a membership function that gives value
of degree of membership x to fuzzy set A, which is:
:U →[0,1] A μ (2)
In a fuzzy set, there are several membership
functions of a new fuzzy set, which result from basic
operation of the fuzzy set, i.e.:
Intersection: A Ç B = min (mA[x], mB[y]) (3)
Union: A È B = max (mA[x], mB[y]) (4)
Complement: ~ A = 1 – mA[x] (5)
Membership function is stated as follows:
0; x ≤ a or x ≥ c
μ (x) = (b – a) / (x – a); a ≤ x ≤ b (6)
(b – x) / (c – b); b ≤ x ≤ c
In a fuzzy system, there is a linguistic variable. This
is a variable that has a value in verbal form in a natural
language. Each linguistic variable is related to a certain
membership function. Figure 2 gives an example of
membership function.
Fuzzy-Eckenrode. The Eckenrode method was
initially known as a weighting multiple criteria method,
which was introduced by Robert T. Eckenrode from
Dunlop and Association, Inc. in 1965 and has been
widely used until today [5–8]. The Eckenrode method
is simpler and more efficient in determining the
importance weight in a decision [9–11]. The Eckenrode
weighting analysis method is one of weighing methods
used in determining the degree of importance, or Weight
(B), from each Criteria (K) and Sub-criteria (SK), which
have been set in decision making [12]. This weight
determination is perceived as very important because
it affects the final total value of each chosen decision.
The concept used in this weighting method is by doing
a change of order to value where, for instance, first
order (1) has the highest rate (value) and the fifth order
(5) has the lowest rate.
Fuzzy-TOPSIS. TOPSIS belongs to the Multiple
Attribute Decision Making (MADM), which was firstly
introduced by Yoon, Yoon et al. and Hwang et al. [13–15].
It has been widely applied in various studies related to
decision making, such as Kumar et al., Han et al., Tyagi,
Estrella et. al., Roszkowska et al., Selim et al. [16–21].
TOPSIS can only be implemented for a criterion whose
weight has been known or calculated before, because
there is a step in TOPSIS which involves the process
of multiplication of criterion weight and the alternative
value of the criterion.
In many situations, the data available is insufficient
for a real life problem, because human assessment,
which is considered as preference, is unclear, and the
preference cannot be estimated with exact numeric
value. The verbal expression, e.g. ‘low’, ‘medium’,
‘high’, etc., is considered as a representation of the
decision maker. Thus, fuzzy logic is necessary in
Figure 1 Cucumis melo packaged in banana stem Figure 2 Membership function
341
Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347
making a structured decision of the preference maker.
The Fuzzy theory helps to measure the uncertainty
associated with human judgement, which is subjective.
Therefore, evaluation is necessary to be done in an
environment. According to Ningrum et al. and Fadhil et
al., fuzzy logic can help improve failure, which happens
when only Eckenrode or TOPSIS method is used [4, 22].
STUDY OBJECTS AND METHODS
This study used Cucumis melo (L.) which was
harvested in two months after planting. The harvested
Cucumis melo was cleaned by washing and then stored
under three different conditions: without banana
stem packaging (A1), with one layer of banana stem
packaging (A2), and with two layers of banana stem
packaging (A3). Cucumis melo was then stored for six
days under three temperature regimes: 10°C (B1), 14°C
(B2), and at room temperature (27–30°C) (B3).
Procedure of assessment. The multi-criteria sensory
assessment of Cucumis melo included aroma, colour,
taste, texture, and overall acceptance (Table 1). The
attribute weight of respondents’ assessment toward the
multi-criteria was determined according to the hedonic
scale. The hedonic scale is a preference of respondent’s
opinion based on likes or dislikes that are converted into
number (Table 2).
The framework of this study included four steps: (1)
selection of respondents and criteria, (2) determination
of criterion weight of the assessment by using the
fuzzy-Eckenrode method, (3) determination of the best
alternative of all treatments by using fuzzy-TOPSIS,
and (4) recommendation of the best acceptance from all
respondents. Figure 3 shows the complete framework.
Combinations of storage conditions were as follows:
A1B1: without banana stem-packaging at 10°C;
A1B2: without banana stem-packaging at 14°C;
A1B3: without banana stem-packaging at 27–30°C;
A2B1: with one layer of banana stem-packaging at 10°C;
A2B2: with one layer of banana stem-packaging at 14°C;
A2B3: with one layer of banana stem-packaging at
27–30°C;
A3B1: with two layers of banana stem-packaging at
10°C;
A3B2: with two layers of banana stem-packaging at
14°C;
A3B3: with two layers of banana stem-packaging at
27–30°C.
Fuzzy-Eckenrode method. According to the
Eckenrode weight calculation method, the respondents
were asked to make a rating (e.g. from R1 until Rn,
where n ranking, j = 1, 2, 3,…, n, ranking j = Rj) for
each criterion (criterion i is notated with Ki, which is
presented in a number of n criteria, i = 1, 2, 3,…, n) [11].
Table 3 shows the obtained data. Next, Ni was calculated
based on Pij and Rn-j.
Rj = ranking order at j, j = 1, 2, 3,…, n
Ki = criterion type i, i = 1, 2, 3,…, n
Pij = number of respondents who chose ranking j for
criterion i
Rn-j = multiplier factor j, which was obtained from the
reduction of number of criteria or number of ranking
(which is n) with the rank order on the column. For
instance, if there are five criteria, so the multiplier factor
for column of 3rd rank (if j = 3) is n–j = 5–3 = 2
Bi = weight of criterion i.
Ni = Gj=1 Prij x Rn-j, j = 1, 2, 3,…, n. (7)
Total Score = Gi=1 Ni, i = 1, 2, 3,…, n. (8)
Table 1 Attributes of multi-criteria sensory assessment
of Cucumis melo
Attribute Assessment consideration
Aroma (C1) Typical, no sour smell
Colour (C2) Yellowish-green
Taste (C3) Sweet and not sour
Texture (C4) Solid, not watery, no wrinkles
Overall acceptance (C5) Yellowish-green in colour, solid,
and sweet
Table 2 Assessment of preference according to hedonic scale
Score Preference
5 Like very much
4 Like
3 Neither like nor dislike
2 Dislike
1 Dislike very much
Figure 3 Research framework
Tabel 3 Calculation of criterion weight according
to the Eckenrode method
Criteria Rank Score Weight
R1 R2 ...... Rj ...... Rn
K1 P11 P12 ...... P1n N1 B1
K2 P21 P22 ...... P2n N2 B2
...... ...... ...... ...... ...... ...... ......
Ki Pij
......
Kn Pn1 Pn2 ...... Pmn Nn Bn
Multiplier
factor
Rn-1 Rn-2 ...... Rn-j Rn-n Total
Score
1.00
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Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347
Then, criterion weight Bi (which are B1, B2,
B3,…, Bn) was calculated, where i = 1, 2, 3,…, 3, by
using the following formula:
Bi = (Ni/Total Score) (9)
To find the level of importance of each sub-criterion
within a criterion, the respondents were also asked
to rank each sub-criterion within a criterion. Then,
by using the same procedure, the weight of each subcriterion
was calculated (B1i
, the weight of sub-criterion
1 in criterion i). Thus, the weighted weight (BT) from
sub-criterion 1 in criterion i was obtained, which was
BT1 = B1i·Bi. Then, to find the score of each criterion, the
respondents were asked to rate each sub-criterion within
each criterion [23].
The assessment of each sub-criterion was calculated
by using a geometric mean formula according to the
assessment result from all respondents, which was
multiplied with the weighted weight of each subcriterion.
Each criterion (K1 to K5) was calculated by
summing up the total score of all sub-criteria in each
criterion. To assess the weighting by the respondents, the
fuzzy-Eckenrode method was applied with the value of
preference, as shown in Table 4.
Fuzzy-TOPSIS method. The analysis with the
fuzzy-TOPSIS method included the following tasks [24]:
To rank the fuzzy from each decision made, Dk;
(k = 1, 2, 3,…, k) can be represented as triangular
fuzzy number ˜Rk; (k = 1, 2, 3,…, K) with membership
function μ˜R (x).
To produce an appropriate alternative, to determine
the criteria of evaluation, and to organise the group
of decision-maker. It was assumed that there were m
alternatives, n criteria of evaluation, and decision k.
To choose the linguistic variable according to the
weight of criterion importance =
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
and
alternative linguistic rankings on criterion (𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
) in
Triangular Fuzzy Number (TFN).
To do a weight aggregation of each criterion to
obtain fuzzy weight aggregate (
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
) in criterion Cj and to
determine the fuzzy aggregate value from alternative Ai
on each criterion Cj.
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑥𝑥̃… 𝑥𝑥̃], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
(10)
i = 1, 2,.., m; and j = 1, 2,..., n
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣𝑗𝑗
+= max {𝑣𝑣𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚
𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
(11)
j = 1, 2,..., n
To build a fuzzy decision matrix.
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚
1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖
𝑖𝑖
𝑗𝑗
𝑢𝑢𝑖𝑖𝑖𝑖
)
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
,
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
(12)
To do normalisation of the decision matrix, where:
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃ 𝑤𝑤𝑖𝑖𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
i = 1, 2,..., m; and j = 1, 2,..., n (13)
Calculating
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
can be done with:
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
(14)
where
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
̃21 𝑥𝑥̃22 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
= max
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
.
To determine the weight normalisation of the
fuzzy decision matrix. Based on different importance
on each criterion, the fuzzy decision of the weighted
normalisation matrix can be arranged as:
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
with, i = 1, 2,..., m; and j = 1, 2,..., n (15)
where:
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, ̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
with, i = 1, 2,..., m; and j = 1, 2,..., n (16)
To determine fuzzy positive ideal solution (FPIS) S+
and fuzzy negative ideal solution (FNIS) S-:
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖
𝑖𝑖 = ̃ 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
(17)
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
(18)
where:
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , ̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
= max
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, ̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖
, 𝑢𝑢𝑗𝑗
∗ )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, ̃2
… . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, ̃2
… . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖𝑢𝑢)
and
𝑥𝑥̃𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖]𝑚𝑚𝑟𝑟̃𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖]𝑚𝑚𝑣𝑣̃𝑖𝑖 = 𝑟𝑟̃𝑖𝑖 𝑤𝑤̃𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, ̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖)
= min
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚
𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚 𝑚𝑚𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚
𝑚𝑚𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , ̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
with
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚 𝑚𝑚 𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚
𝑚𝑚
𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = 𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
are
TFN normalisation weight.
To calculate the interval between each alternative
value and the value of FPIS (Fuzzy Positive Ideal
Solution) and FNIS (fuzzy negative ideal solution).
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖
= 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
(19)
𝑖𝑖𝑖𝑖 1
𝑘𝑘 𝑖𝑖𝑖𝑖
1 𝑖𝑖𝑖𝑖
2 𝑖𝑖𝑖𝑖
𝑘𝑘 𝑗𝑗𝑗𝑗
1 𝑗𝑗
2 𝑗𝑗
𝑘𝑘1 2 𝑛𝑛
𝐷𝐷 ̃
=
1
2
𝑚𝑚
𝑛𝑛
𝑛𝑛
𝑚𝑚1 𝑚𝑚2 𝑚𝑚𝑚𝑚
𝑊𝑊 ̃
𝑛𝑛𝑅𝑅 ̃𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 ̃𝑖𝑖𝑖𝑖 𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ 𝑗𝑗
max 𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚
𝑖𝑖𝑖𝑖 ̃𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
+ 𝑛𝑛
− 𝑛𝑛
𝑗𝑗
max 𝑖𝑖𝑖𝑖and 𝑗𝑗
min 𝑖𝑖𝑖𝑖with 𝑗𝑗
= 1
3 1 2 1 2 2
1
+ 𝑖𝑖𝑖𝑖𝑗𝑗
𝑛𝑛 𝑗𝑗
=1
1
− 𝑖𝑖𝑖𝑖𝑗𝑗
𝑛𝑛 𝑗𝑗
=1
𝑖𝑖 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− (𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
(20)
𝑥𝑥̃𝑖𝑖𝑖𝑖 = 1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + … . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃= [𝑟𝑟̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)2
𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
(21)
To calculate the closeness coefficient (CCi) and the
ranking according to the coefficient value obtained using
the following equation:
𝑥𝑥̃𝑖𝑖 =
1
𝑘𝑘 [𝑥𝑥̃𝑖𝑖𝑖𝑖
1 + 𝑥𝑥̃𝑖𝑖𝑖𝑖
2 + . + 𝑥𝑥̃𝑖𝑖𝑖𝑖
𝑘𝑘 ]
𝑤𝑤̃𝑗𝑗=
1
𝑘𝑘
[𝑤𝑤̃𝑗𝑗
1 + 𝑤𝑤̃𝑗𝑗
2 + … . + 𝑤𝑤𝑗𝑗
𝑘𝑘]
𝐶𝐶1 𝐶𝐶2 … 𝐶𝐶𝑛𝑛
𝐷𝐷 ̃
=
𝐴𝐴1
𝐴𝐴2
𝐴𝐴𝑚𝑚
[
𝑥𝑥̃11 𝑥𝑥̃12 … 𝑥𝑥̃1𝑛𝑛
𝑥𝑥̃21 𝑥𝑥̃22 … 𝑥𝑥̃2𝑛𝑛
𝑥𝑥̃𝑚𝑚1 𝑥𝑥̃𝑚𝑚2 … 𝑥𝑥̃𝑚𝑚𝑚𝑚
], 𝑊𝑊 ̃
= [𝑤𝑤̃1, 𝑤𝑤̃2 … 𝑤𝑤̃𝑛𝑛]
𝑅𝑅 ̃
= [𝑟𝑟̃𝑖𝑖 𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚 𝑟𝑟̃𝑖𝑖𝑖𝑖 = (𝑙𝑙𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑚𝑚𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , 𝑢𝑢𝑖𝑖𝑖𝑖
𝑈𝑈𝑗𝑗
∗ , )
𝑈𝑈𝑗𝑗
∗= max 𝑢𝑢𝑖𝑖𝑖𝑖 𝑉𝑉 ̃
= [𝑣𝑣̃𝑖𝑖𝑖𝑖]𝑚𝑚𝑚𝑚𝑚𝑚
𝑣𝑣̃𝑖𝑖𝑖𝑖 = 𝑟𝑟̃𝑖𝑖𝑖𝑖 𝑤𝑤̃𝑖𝑖𝑖𝑖
𝑆𝑆+ = (𝑣𝑣̃1
+, 𝑣𝑣̃2
+, … . , 𝑣𝑣̃𝑛𝑛
+)
𝑆𝑆− = (𝑣𝑣̃1
−, 𝑣𝑣̃2
−, … . , 𝑣𝑣̃𝑛𝑛
−)
𝑣𝑣̃𝑗𝑗
+= max {𝑣𝑣𝑖𝑖𝑖𝑖3} and 𝑣𝑣̃𝑗𝑗
−= min {𝑣𝑣𝑖𝑖𝑖𝑖1} with 𝑣𝑣̃𝑗𝑗
𝑑𝑑 (𝐴𝐴1, 𝐴𝐴2) = √1
3 [(𝑙𝑙1 − 𝑙𝑙2)2 + (𝑚𝑚1 − 𝑚𝑚𝑚𝑚2)2 + (𝑢𝑢 − 𝑢𝑢2)𝑑𝑑1
+ = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
+), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝑑𝑑1
− = Σ 𝑑𝑑(𝑣𝑣̃𝑖𝑖𝑖𝑖, 𝑣𝑣̃𝑗𝑗
−), 𝑖𝑖 = 1, 2,…,𝑚𝑚 𝑛𝑛 𝑗𝑗
=1
𝐶𝐶𝐶𝐶𝑖𝑖 = 𝑑𝑑𝑖𝑖
−
𝑑𝑑𝑖𝑖
++𝑑𝑑𝑖𝑖
− , 𝑖𝑖 = 1, 2, … . , 𝑚𝑚
(𝑤𝑤̃𝑗𝑗= 𝑙𝑙𝑖𝑖𝑖𝑖, 𝑚𝑚𝑖𝑖𝑖𝑖, 𝑢𝑢𝑖𝑖𝑖𝑖)
(22)
Table 4 Scale of weighting comparison among criteria
of fuzzy-Eckenrode method
Scale Annotation TFN membership function
~1 Very unimportant 1, 1, 2
~2 Less important 1, 2, 3
~3 Neutral 2, 3, 4
~4 Important 3, 4, 5
~5 Very important 4, 5, 5
343
Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347
To rate each alternative by the respondents, we used
the fuzzy-TOPSIS method with preference value, as in
Table 5.
Figure 4 illustrates the procedure of the analysis.
Selection of respondents. A total of 10 respondents
were chosen to do a multi-criteria sensory assessment of
Cucumis melo. The respondents were selected according
to several criteria. The potential respondents had to:
1. like Cucumis melo, raw or processed;
2. be experienced in sensory assessment;
3. be healthy, as flu, cough, mouth ulcers, etc. can
bother the sensory assessment process;
4. be able to distinguish colours.
RESULTS AND DISCUSSION
Determination of assessment criteria weight.
A hedonic scale was used to evaluate the results of
determination of respondents’ assessment of criteria
weight towards multi-criteria which were considered in
the sensory assessment. After that, they were translated
into fuzzy logic functions (Table 6).
As for the data of respondents’ assessment towards
criteria of importance weight determination from
each sensory attribute, the values of lower bound
(low), middle (medium), and upper bound (upper) were
arranged as summarised on Table 7. The next step was
to calculate the score and the weight of each criterion.
Figure 5 represents a radar diagram.
According to the respondents’ assessment of the
criteria with the help of the fuzzy-Eckenrode method,
the order of criteria weight was obtained from the
highest to the lowest: (1) overall acceptance, 0.216; (2)
colour, 0.211; (3) aroma, 0.203; (4) taste, 0.191; and (5)
texture 0.176.
Determination of the best alternative. The
priority of the best alternative from the multi-criteria
sensory assessment of Cucumis melo was determined
by summarising all respondents’ preferences. The
preferences were chosen based on the mode number,
i.e. the value that appears most often from each choice
of material treatment. The mode number was chosen
by the respondents. The next step was to arrange the
matrix of the respondents’ assessment on all alternatives
(Table 8). The data of respondents’ assessment was then
transformed into TFN linguistic data, as presented in
Table 9.
After that, we formulated the normalised weight
matrix on each alternative. The value normalisation can
be done by using Eqs. (13) and (14). Table 10 shows the
results of the TFN value normalisation.
Table 5 Comparison scale of determination
of the fuzzy-TOPSIS method alternative
Scale TFN Linguistics
Dislike very much (STS) 1, 1, 2
Dislike (TS) 1, 2, 3
Neither like nor dislike (N) 2, 3, 4
Like (S) 3, 4, 5
Like very much (SS) 4, 5, 5
Figure 4 Steps of the fuzzy-TOPSIS method analysis
Table 6 Respondents’ weighting score of criteria based on the
fuzzy-Eckenrode method
No Criteria Order
1 2 3 4 5
1 C1 ~4 ~3 ~1 ~1 ~1
2 C2 ~3 ~4 ~1 ~1 ~1
3 C3 ~3 ~1 ~4 ~1 ~1
4 C4 ~2 ~2 ~4 ~1 ~1
5 C5 ~5 ~2 ~1 ~1 ~1
Nilai
(Gcriteria-order)
4 3 2 1 0
Table 7 TFN value of experts’ weighting on criteria of the fuzzy-Eckenrode method
No Criteria 1 2 3 4 5 Score Weight
l m u l m u l m u l m u l m u
1 C1 4 5 5 1 2 3 1 1 2 1 1 2 1 1 2 82 0.206
2 C2 3 4 5 1 2 3 1 2 3 1 1 2 1 1 2 84 0.211
3 C3 1 2 3 1 1 2 3 4 5 1 2 3 1 1 2 76 0.191
4 C4 1 1 2 1 2 3 3 4 5 1 2 3 1 1 2 70 0.176
5 C5 3 4 5 1 2 3 1 1 2 1 2 3 1 1 2 86 0.216
Nilai (Gcriteria-order) 4 3 2 1 0 398 1,000
l = lower, m = middle, u = upper
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Table 8 Matrix of experts’ assessment on alternatives
Alternatives Criteria
C1 C2 C3 C4 C5
A1B1 3 2 1 1 3
A1B2 3 2 2 1 3
A1B3 2 2 2 1 1
A2B1 4 3 3 2 4
A2B2 4 4 3 2 4
A2B3 2 2 2 1 1
A3B1 5 4 5 4 5
A3B2 5 5 5 5 5
A3B3 1 1 1 1 1
Then, we arranged the matrix of multiplication
between criteria weights and normalisation value of each
alternative. This process can be done by using Eqs. (15)
and (16). Table 11 summarises the results of the matrix
multiplication.
The next step was to determine the positive ideal
solution value (FPIS) S+ and the negative ideal solution
value (FNIS) S–. When determining both values, the
characteristic of data available should be taken into
consideration. To obtain both groups of values, one can
use Eqs. (17) and (18). Table 12 demonstrates FPIS and
FNIS values.
After that, the interval between each alternative
value and FPIS and FNIS was calculated by using
Eqs. (19), (20), and (21). The results of the interval
calculation between alternative value toward FPIS and
FNIS can be observed from Table 13 and Table 14.
We evaluated the criteria distance value to the fuzzy
positive ideal solution (FPIS) and the fuzzy negative ideal
solution (FNIS) according to comparison of d+ and d–.
It showed preference of product acceptance on a radar
diagram (Fig. 6). For instance, the treatment of Cucumis
Figure 5 Radar diagram of criteria weight
melo without packaging at temperature of 10°C (A1B1)
had such d+ and d– values that showed the biggest
distance from the positive ideal and the negative ideal.
The final step was to calculate the closeness
coefficient (CCi) of each alternative by using
Eq. (22). From the calculation result, we obtained
ranking from the highest to the lowest (Table 15). The
biggest coefficient value was the main alternative, which
was suggested to be chosen or prioritised, compared
to other alternatives based on respondents’ preference
(product acceptance).
According to the closeness coefficient (CCi), an
alternative ranking can be arranged from the biggest to
the lowest as follows: two-layer banana stem-packaging
at 14°C (A3B2), two-layer banana stem-packaging at
10°C (A3B1), one-layer banana stem-packaging at 14°C
(A2B2), one-layer banana stem-packaging at 10°C
(A2B1), without banana stem packaging at 14°C (A1B2),
one-layer banana stem-packaging at room temperature
(A2B3), without banana stem packaging at 10°C (A1B1),
without banana stem packaging at room temperature
Table 9 Matrix of respondents’ assessment on alternative in TFN scale
Alternatives Criteria
Aroma (0.191,
0.206, 0.211)
Colour (0.206,
0.211, 0.216)
Taste (0.176,
0.191, 0.206)
Texture (0.176,
0.176, 0.191)
Overall acceptance
(0.204, 0.216, 0.216)
A1B1 (2, 3, 4) (1, 2, 3) (1, 1, 2) (1, 1, 2) (2, 3, 4)
A1B2 (2, 3, 4) (1, 2, 3) (1, 2, 3) (1, 1, 2) (2, 3, 4)
A1B3 (1, 2, 3) (1, 2, 3) (1, 2, 3) (1, 1, 2) (1, 1, 2)
A2B1 (3, 4, 5) (2, 3, 4) (2, 3, 4) (1, 2, 3) (3, 4, 5)
A2B2 (3, 4, 5) (3, 4, 5) (2, 3, 4) (1, 2, 3) (3, 4, 5)
A2B3 (1, 2, 3) (1, 2, 3) (1, 2, 3) (1, 1, 2) (1, 1, 2)
A3B1 (4, 5, 5) (3, 4, 5) (4, 5, 5) (3, 4, 5) (4, 5, 5)
A3B2 (4, 5, 5) (4, 5, 5) (4, 5, 5) (4, 5, 5) (4, 5, 5)
A3B3 (1, 1, 2) (1, 1, 2) (1, 1, 2) (1, 1, 2) (1, 1, 2)
A1B1: without banana stem-packaging at 10°C
A1B2: without banana stem-packaging at 14°C
A1B3: without banana stem-packaging at 27–30°C
A2B1: with one layer of banana stem-packaging at 10°C
A2B2: with one layer of banana stem-packaging at 14°C
A2B3: with one layer of banana stem-packaging at 27–30°C
A3B1: with two layers of banana stem-packaging at 10°C
A3B2: with two layers of banana stem-packaging at 14°C
A3B3: with two layers of banana stem-packaging at 27–30°C
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Table 10 Matrix of TFN scale normalisation
Alternative Criteria
Aroma
(0.191, 0.206, 0.211)
Colour
(0.206, 0.211, 0.216)
Taste
(0.176, 0.191, 0.206)
Texture
(0.176, 0.176, 0.191)
Overall acceptance
(0.204, 0.216, 0.216)
A1B1 (0.40, 0.60, 0.80) (0.20, 0.40, 0.60) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40) (0.40, 0.60, 0.80)
A1B2 (0.40, 0.60, 0.80) (0.20, 0.40, 0.60) (0.20, 0.40, 0.60) (0.20, 0.20, 0.40) (0.40, 0.60, 0.80)
A1B3 (0.20, 0.40, 0.60) (0.20, 0.40, 0.60) (0.20, 0.40, 0.60) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40)
A2B1 (0.60, 0.80, 1.00) (0.40, 0.60, 0.80) (0.40, 0.60, 0.80) (0.20, 0.40, 0.60) (0.60, 0.80, 1.00)
A2B2 (0.60, 0.80, 1.00) (0.60, 0.80, 1.00) (0.40, 0.60, 0.80) (0.20, 0.40, 0.60) (0.60, 0.80, 1.00)
A2B3 (0.20, 0.40, 0.60) (0.20, 0.40, 0.60) (0.20, 0.40, 0.60) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40)
A3B1 (0.80, 1.00, 1.00) (0.60, 0.80, 1.00) (0.80, 1.00, 1.00) (0.60, 0.80, 1.00) (0.80, 1.00, 1.00)
A3B2 (0.80, 1.00, 1.00) (0.80, 1.00, 1.00) (0.80, 1.00, 1.00) (0.80, 1.00, 1.00) (0.80, 1.00, 1.00)
A3B3 (0.20, 0.20, 0.40) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40) (0.20, 0.20, 0.40)
Table 11 Matrix of multiplication of criteria weights and alternative normalisation values
Alternatives Criteria
Aroma
(0.203, 0.204, 0.209)
Colour
(0.196, 0.203, 0.204)
Taste
(0.188, 0.196, 0.203)
Texture
(0.188, 0.188, 0.196)
Overall acceptance
(0.204, 0.209, 0.209)
A1B1 (0.08, 0.12, 0.17) (0.04, 0.08, 0.13) (0.04, 0.04, 0.08) (0.04, 0.04, 0.08) (0.08, 0.13, 0.17)
A1B2 (0.08, 0.12, 0.17) (0.04, 0.08, 0.13) (0.04, 0.08, 0.12) (0.04, 0.04, 0.08) (0.08, 0.13, 0.17)
A1B3 (0.04, 0.08, 0.13) (0.04, 0.08, 0.13) (0.04, 0.08, 0.12) (0.04, 0.04, 0.08) (0.04, 0.09, 0.13)
A2B1 (0.11, 0.16, 0.21) (0.08, 0.13, 0.17) (0.07, 0.11, 0.16) (0.04, 0.07, 0.11) (0.12, 0.17, 0.22)
A2B2 (0.11, 0.16, 0.21) (0.12, 0.17, 0.22) (0.07, 0.11, 0.16) (0.04, 0.07, 0.11) (0.12, 0.17, 0.22)
A2B3 (0.04, 0.08, 0.13) (0.04, 0.08, 0.13) (0.04, 0.08, 0.12) (0.04, 0.04, 0.08) (0.04, 0.04, 0.09)
A3B1 (0.15, 0.21, 0.21) (0.12, 0.17, 0.22) (0.14, 0.19, 0.21) (0.11, 0.14, 0.19) (0.16, 0.22, 0.22)
A3B2 (0.15, 0.21, 0.21) (0.16, 0.21, 0.22) (0.14, 0.19, 0.21) (0.14, 0.18, 0.19) (0.16, 0.22, 0.22)
A3B3 (0.04, 0.04, 0.08) (0.04, 0.04, 0.09) (0.04, 0.04, 0.08) (0.04, 0.04, 0.08) (0.04, 0.04, 0.09)
Table 12 Positive ideal solution and negative ideal solution values
Criteria Aroma Colour Taste Texture Overall acceptance
S (+) (0.21, 0.21, 0.21) (0.22, 0.22, 0.22) (0.21, 0.21, 0.21) (0.19, 0.19, 0.19) (0.22, 0.22, 0.22)
S (–) (0.04, 0.04, 0.04) (0.04, 0.04, 0.04) (0.04, 0.04, 0.04) (0.19, 0.19, 0.19) (0.22, 0.21, 0.22)
Table 13 Intervals between criteria value and FPIS
FPIS
(d+)
Criteria d+
Aroma Colour Taste Texture Overall
acceptance
A1B1 0.096 0.136 0.156 0.143 0.095 0.626
A1B2 0.096 0.136 0.133 0.143 0.095 0.603
A1B3 0.134 0.136 0.133 0.143 0.135 0.681
A2B1 0.062 0.096 0.097 0.122 0.059 0.436
A2B2 0.062 0.060 0.097 0.122 0.059 0.400
A2B3 0.134 0.136 0.133 0.143 0.160 0.706
A3B1 0.034 0.060 0.039 0.057 0.030 0.219
A3B2 0.034 0.030 0.039 0.030 0.030 0.162
A3B3 0.158 0.161 0.156 0.143 0.160 0.778
Table 14 Interval between criteria value and FNIS
FPIS
(d-)
Criteria d-
Aroma Colour Taste Texture Overall
acceptance
A1B1 0.093 0.057 0.027 0.143 0.095 0.416
A1B2 0.093 0.057 0.056 0.143 0.095 0.445
A1B3 0.057 0.057 0.056 0.143 0.135 0.449
A2B1 0.131 0.094 0.090 0.122 0.059 0.496
A2B2 0.131 0.134 0.090 0.122 0.059 0.536
A2B3 0.057 0.057 0.056 0.143 0.160 0.474
A3B1 0.154 0.134 0.147 0.057 0.030 0.521
A3B2 0.154 0.158 0.147 0.030 0.030 0.518
A3B3 0.027 0.026 0.027 0.143 0.160 0.384
(A1B3), and two-layer banana stem-packaging at room
temperature (A3B3) (Fig. 7).
The analysis with fuzzy-TOPSIS approach showed
that the respondents preferred Cucumis melo stored
in a two-layer banana stem packaging at 14°C (A3B2).
Since the scores were fairly close between Cucumis melo
stored in a two-layer banana stem packaging at 14°C
(A3B2) and Cucumis melo stored in a two-layer banana
stem packaging at 10°C (A3B1), both products were
favored by consumers (respondents’ preferences).
CONCLUSION
According to the consumer assessment of all types
of the six-day storage of Cucumis melo, the optimal
346
Fadhil R. et al. Foods and Raw Materials, 2019, vol. 7, no. 2, pp. 339–347
storage conditions involved packaging with two layers
of banana stem at the temperature of 14°C (A3B2). The
fuzzy-Eckenrode and fuzzy-TOPSIS methods were very
helpful in calculating the results of the multi-criteria
sensory assessment through weighing. They made the
Figure 6 Evaluation of d+ and d–
Figure 7 Alternative ranking of Cucumis melo product
acceptance
process of determining consumers’ acceptance easier,
faster, and more certain.
CONFLICT OF INTEREST
The authors declare no conflict of interest.
ACKNOWLEDGEMENTS
We thank our laboratory assistants at the Post-
Harvest Engineering Laboratory, Faculty of Agriculture,
Syiah Kuala University, especially Riza Rahmah, for
their support and assistance in organising respondents
and material preparation for this research.

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