Available via license: CC BY

Content may be subject to copyright.

Email: odugodwin@gmail.com

PRINT ISSN 1119-8362

Electronic ISSN 1119-8362

J. Appl. Sci. Environ. Manage.

Vol. 23 (8) 1449-1457 August 2019

Full-text Available Online at

https://www.ajol.info/index.php/jasem

http://ww.bioline.org.br/ja

Weighting Methods for Multi-Criteria Decision Making Technique

ODU, G.O.

Department of Mechanical Engineering, Faculty of Engineering, Delta State University, Abraka, Oleh Campus, 331107, Nigeria.

Email: odugodwin@gmail.com

ABSTRACT

:

Determining criteria weights is a problem that arises frequently in many multi-criteria decision-making

(MCDM) techniques. Taking into account the fact that the weights of criteria can significantly influence the outcome

of the decision-making process, it is important to pay particular attention to the objectivity factors of criteria weights.

This paper provides an overview of different weighting methods applicable to multi-criteria optimization techniques.

There are a lot of concept been reported from the literature that are very useful in solving multicriteria problems. The

present work emphasized on the use of these weighting methods in determining the criteria preference of each criterion

to bring about desirable properties and in order to establish and satisfy a multiple measure of performance across all

the criteria selected by identifying the best options possible. And from the results, it shows that subjective weighting

methods are easy and straight forward in terms of their computations than the objective weighting methods which

derived their information from each criterion by adopting a mathematical function to determine the weights without

the decision-maker’s input,. This can be seen from the pairwise comparison which gives an internal storage and random

access memory of a smart phone a weight value of 0.33 and 0.22 respectively as they have the highest criteria weights.

DOI: https://dx.doi.org/10.4314/jasem.v23i8.7

Copyright: Copyright © 2019 Odu. This is an open access article distributed under the Creative Commons

Attribution License (CCL), which permits unrestricted use, distribution, and reproduction in any medium,

provided the original work is properly cited.

Dates: Received: 20 May 2019; Revised: 27 July 2019; Accepted 31 July 2019

KEYWORDS: Multi-criteria, Decision-making, Relative importance, Alternative, Criteria

In most multi-criteria decision making (MCDM)

models, assigning weights to criteria is an important

step that needs to be reexamined. Though, determining

the weights of criteria is one of the key problems that

arise in multi-criteria decision making (Dragan et al.,

2018). There are various weighting methods that have

been proposed in literature and applied for solving

different MCDM problems such as goal programming,

Analytic Hierarchy Process (AHP), weighted score

method, VIKOR, TOPSIS, etc. These weighting

methods are classified in different ways: Direct criteria

weighting methods (scaling, ranking-weight, point

allocation procedures and an indirect approach

(weight derived from theories and mathematical

model). In practice, it is difficult even for a single

decision maker to supply numerical relative weights of

different decision criteria. Naturally, obtaining criteria

weights from several decision makers is more

difficult. Quite often, decision makers are much more

comfortable in simply assigning ordinary ranks to the

different criteria under consideration. In such cases,

relative criteria weights can be derived from criteria

ranks supplied by decision makers. The decision for

selecting an appropriate weighting method is a difficult

task in solving a multi-criteria decision problem.

Several researchers have dismissed the difficulty in

measuring the criteria weights and assume that the

importance of criteria weights is conversant with all

decision makers (Zardari et al., 2015). However, the

validity of criteria weights obtained from different

weighting methods cannot be ignored so as not to avoid

any misuse of the MCDM models and getting reliable

model results. MCDM methods can help to improve

the quality of decisions by making the decision

making process more explicit, rational, and efficient

(Arvind and Janpriy, 2018). The author pointed out

that multi criteria decision making (MCDM) is

regarded as a main part of modern decision science

and operational research, which contains multiple

decision criteria and multiple decision alternatives.

Several researchers have come up with different

methods of determining the criteria weights of a multi-

criteria decision making problem (Ginevicius and

Podvezko 2005; Diakoulaki et al, 1995, Aldian and

Taylor 2005; Dragan et al., 2018). Amongst is the

weighted sum method (WSM) known to be the earliest

and probably the most widely used method. The WSM

was later modified to weighted product method (WPM)

in order to overcome some gaps associated with it. In

1977, Saaty proposed the analytical hierarchy process

(AHP) and it has recently become one of the popular

methods in most MCDM techniques. Nowadays,

modification to the AHP is considered to be prevalent

than the original approach, e.g., the fuzzy AHP method.

However, some challenges surrounds the theoretical

Weighting Methods for Multi-Criteria….. 1450

ODU, GO

basis of the method, it is easy to use and gives results

that are expected to the users. Despite its ease of use, the

procedure for processing information obtained from the

decision maker is difficult to ascertain. This makes the

method less suitable for situations with many

stakeholders. Moreover for AHP, the number of

pairwise comparisons increases rapidly with the number

of criteria which makes it cumbersome. Other

commonly used methods are the ELETRE, VIKOR and

the TOPSIS methods. The ELETRE (Elimination and

Choice Translating Reality) was first introduced in 1968

by Bernard Roy to deal with outranking relations which

deals with problem of ranking alternatives from the best

to worst by using pairwise comparisons among

alternatives considering each criterion separately.

Though with the outranking relationship, decision

maker may still take the risk of regarding one of the

alternatives better than the other. This means that the

decision maker has a weak or strict preference for one

of the alternatives and sometimes unable to identify the

most preferred alternatives because of their difficulty to

determine the alternative over the other. However, this

method has the ability of eliminating less favourable

alternatives and is convenient when there are decision

problems that require fewer criteria with a large number

of alternatives. In addition, ELECTRE method consists

of a pairwise comparison of alternatives, based on the

degree to which evaluations of the alternatives and the

preference weights confirm or contradict the pairwise

dominance relationship between alternatives. It

examines both the degree to which the preference

weights are in agreement with pairwise dominance

relationships and the degree to which weighted

evaluations differ from each other. These stages are

based on a ‘‘concordance and discordance’’ set;

hence, this method is also called concordance analysis.

The VIKOR method is a multi-criteria decision

making (MCDM) method. It was originally developed

by Serafim Opricovic to solve decision problems with

conflicting and non-commensurable criteria, assuming

that compromise is acceptable for conflict resolution

(Arvind and Janpriy, 2018). This method focuses on

ranking and selecting from a set of alternatives and

determines the compromise solution closest to the

ideal solution. Chatterjee et al. (2012) proposed

decision-making methodology for material selection

using compromise ranking method known as Vlse

Kriterijumska Optimizacija Kompromisno Resenje’

(VIKOR), which means multi-criteria optimization

and compromise solution. The TOPSIS method is

based on technique of ranking preferences by

similarity to the ideal solution (TOPSIS) to aid in

material selection process proposed by Hwang and

Yoon in 1980 (Xu, 2007). According to this technique,

the best alternative would be the one that is closest to

the positive-ideal solution and farthest from the

negative ideal solution. The primary concept of

TOPSIS approach is that the most preferred alternative

should not only have the shortest distance from the

positive ideal solution, but also have the farthest

distance from the negative ideal solution (Vinodh et

al., 2014). The Euclidean distance approach was

proposed to evaluate the relative closeness of the

alternatives to the ideal solution. Thus, the order of

preference of the alternatives can be obtained by a

series of comparisons of these relative distances. The

entropy method is the method used for assessing the

weight in a given problem because with this method,

the decision matrix for a set of candidate materials

contains a certain amount of information. The entropy

works based on a predefined decision matrix. Using

the entropy method, it is possible to combine the

material designer’s priorities with that of the

sensitivity analysis. The TOPSIS method first converts

the various criteria dimensions into non-dimensional

criteria. When the designer finds no reason to give

preference to one criterion over another, the principle

of insufficient reason (Star and Greenwood 1977)

suggests that each one should be equally preferred.

However, some modification of TOPSIS has been

proposed by Jahanshahloo (2006), Liu and Zeng,

(2008), Rao and Davim, (2008), and Rao and Patel

(2011).

Weights assigned to criteria in multi-criteria evaluation

has both qualitative and quantitative data so as to make

sure that the weight is taking into account for better and

more accurate decision making. However, assigning

weights using qualitative data to criteria can be

influenced by decision maker preference, and due to this

set back, Saaty (1977) proposed a numerical scale of

“1– 9” in order to transform qualitative data into

quantitative by describing ‘1’ as equal importance and

‘9’ as extreme importance (Abel et al., 2018). Weights

classification can also be grouped into three categories:

Subjective, objective and integrated or combined

weighting approach (Ginevicius and Podvezko 2005).

Subjective weight determination is based on expert

opinion, and in order to get the subjective judgments,

analyst normally presents the decision makers a set of

questions in the process. However, subjective criteria

weight determination is often time consuming

especially when there is no agreement between decision

makers of the problem under consideration. Example of

the subjective weighting method is the Analytical

Hierarchy analysis (AHP), Olson (2008) wrote on the

subjectivity in multiple criteria decision analysis; he

argued that judgment is at the heart of human decision-

making and, therefore, considered judgment to be

subjective. If a decision were to be made objectively,

one should simply adopt the “decision support” view

Weighting Methods for Multi-Criteria….. 1451

ODU, GO

that human decision-makers should be entrusted with

the final decision, and that every model is imperfect.

Models do not include all factors. Even the most

careful attempts at objective measurement will

inevitably involve some inaccuracy. The author also

pointed out that we must accredit our own judgment as

the paramount arbiter. In the objective weighting

methods, criteria weights are derived from information

gathered in each criterion through mathematical

models without any consideration of the decision

maker’s intervention (Aldian and Taylor, 2005). The

integrated weighting approach is a weighting method

based on the combination of subjective weighting and

objective weighting methods. It focuses on the

principle of integrating the subjective weights based

on expert’s opinion due to his/her knowledge and

experience in the relevant field and the information

gathered from the criteria data in a mathematical form

(objective weighting method). In the subsequent

section, the mathematical function and specific

examples of each of these methods will be illustrated

and evaluated.

Now, let us look at some of the most common

subjective weights that have been used in previous

MCDM studies are shown in Table 1.

Table 1: Classification of weighting methods

Weighting methods

Subjective weighting methods Objective weighting methods Integrated weighting methods

Point allocation Entropy method Multiplication synthesis

Direct rating Criteria Importance Through Inter-criteria

Correlation (CRITIC)

Additive synthesis

Ranking method Mean weight Optimal weighting based on sum of

squares

Pairwise comparison (AHP) Standard deviation Optimal weighting based on relational

coefficient of graduation

Ratio method Statistical variance procedure

Swing method Ideal point method

Delphi method

Nominal group technique

Simple Multi-attribute Ranking

Technique (SMART)

Subjective weighting methods: The most commonly

used subjective weighting methods are listed in Table

1 are as follows:

(1) The point allocation method: This is one of the

simplest methods used to determine criteria weights

according to the priority of criteria, a decision-maker

allocates a certain number of points to each criterion.

The more points a criterion receives, the greater its

relative importance (Golaszewski et al., 2012). In this

scenario, the decision maker is asked to allocate 100

points across the criteria under consideration. The total

of all criterion weights must sum up to 100. This

method is easy to normalize. However, the weights

obtained from the use of point allocation method are

not very precise, and the method becomes more

difficult as the number of criteria increases to 6 or

more. For example, consider five key quality

characteristics of smart phone one should look out for:

cost, display resolution, battery life, random memory,

and internal storage.

Table 2: Smart phone criteria weights using point allocation method

S/N Criteria Weights

1 Cost 10

2 Display Resolution 35

3 Battery Life 15

4 Random Access Memory (RAM) 25

5 Internal Storage 15

Total 100

(2) The direct Rating method: The direct rating

method is a type of approach in which the decision-

maker first ranks all the criteria according to their

importance. The rating does not constrain the decision

maker’s responses as the fixed point scoring methods

does. It is possible to alter the importance of one

criterion without adjusting the weight of another

(Arbel, 1989).

(3) The pairwise comparisons: This method is used for

analyzing multiple populations in pairs to determine

whether they are significantly different from one

another. It can also put as a method where the

decision-maker compares each criterion with others

and determines the level of preferences for each pair

of such criteria. The use of ordinal scale (1 - 9) is

adopted to help in determining the preference value of

one criterion against the other. And one of the most

commonly applied methods based on pairwise

comparisons is the Analytical Hierarchy process

(AHP) method. The number of comparisons can be

determined by

2

)1(

nn

c

p

(1)

Weighting Methods for Multi-Criteria….. 1452

ODU, GO

Where

p

c

= the number of comparisons: n = the

number of criteria

Determining the criteria weights based on pairwise

comparisons method has three main steps and can be

implemented as follows. The first step is to develop a

matrix by comparing the criteria as shown in Table 4.

Intensity values are used to fill the matrix, such as

(1,3,5,7,9) representing equal importance, moderate

importance of one over the other, strong importance,

very strong importance, extreme importance

respectively,. While the ordinal scale of 2,4,6 and 8 are

intermediate values or when compromise is needed

and can be represented as follows: equally to

moderately preferred – 2; moderately to strongly

preferred - 4; strongly to very strong importance -6;

and very strong to extremely strong importance -8.

The diagonal in the matrix is always 1 and the lower

left values are inverse values if activity i has one of the

above numbers assigned to it when compared with

activity j, then j has the reciprocal value when

compared with i. To fill the lower triangular matrix,

we use the reciprocal values of the upper diagonal.

Thus we have complete comparison matrix.

The second step is to calculate the criteria weight,

which is also known as priority value or the principal

eigenvector. This is done by using either of the

following methods:

Method 1: By summing the values in each column,

dividing each element by the column total, and

dividing the sum of the normalized scores for each row

by the number of criteria as shown in the given

example (Table 4). The calculation for the priority

value of the first row in the matrix is given as:

summation for the first column total is 22, and the

remaining four columns gives 4.33, 3.44, 5.70, and

5.25. Therefore, the priority value for the first row is

gives

05.05/)

25.5

4/1

70.5

5/1

44.3

9/1

33.4

3/1

22

1

(

(2)

Method 2: Multiplying together the entries in each

row of the matrix and then taking the nth root of that

product gives a very good approximation. The nth

roots are summed and that sum is used to normalize

the eigenvector elements to add to 1.00. For the

example, in Table 4, the number of criteria or attribute

is five, which is the fifth root for the first row is 0.283

and that is divided by 5.88 to give 0.05 as the first

criteria weight.

The third step is to estimate the consistency for

sensitivity analysis known as consistency ratio (CR).

If the consistency ratio is less than 0.1, then the ratio

indicates a reasonable level of consistency in the

pairwise comparisons, but once the CR is greater than

0.1, it shows that the pairwise comparisons are

inconsistent in judgment. Sensitivity analysis can be

useful in providing information as to the robustness of

any decision. In order to compute the consistency

ratio, the following procedure needs to be followed:

(a) multiply each value in the first row of the pairwise

comparisons matrix by corresponding criteria weight

or eigenvector to obtain a new vector. For example,

10.005 +

0.20 +

0.33 +

0.22 +

0.2 = 0.246

(3)

(b) Repeat step (a) for remaining columns, that is, the

remaining four rows give 1.100, 1.840, 1.179, and

1.040 as the five elements of

max

. (c) Divide each

elements of the vector of weighted sums obtained in

step a-b by the corresponding priority value. Each

component of (0.246, 1.100, 1.840, 1.179, 1.040) by

the corresponding criteria weight. This gives

926.4

05.0

246.0

for the first row, other values are

5.500, 5.576, 5.359, and 5.200. (d) Then compute the

average of the values found in step c, let

max

be

the average. The mean of these values is 5.312, which

is the estimate of the

max

. If any of the estimates for

max

turns out to be less than n, or 5 in this case, there

has been an error in the calculation. (e) Compute the

consistency index (CI), which is defined as

1

max

n

n

CI

(4)

(f) Using random judgments from Table 3 which was

derived from Saaty’s book, in which a set of judgments

for the corresponding value from large samples of

matrices for the computation of consistency ratio, in

which the upper row is the order of the random matrix,

and the lower row is the corresponding index of the

consistency for random judgments referred to as

random index

Table 3: Random Index

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

RI 0.00 0.00 0.58 0.90 1.12 1.24 1.32 1.41 1.45 1.49 1.51 1.54 1.56 1.57 1.59

Source: Saaty, (1980)

Weighting Methods for Multi-Criteria….. 1453

ODU, GO

(g) Therefore, the consistency ratio, CR is

RI

CI

CR

; That means

07

.0

12.1

078.0 CR

(5)

Accept the matrix if consistency ratio is less than 0.1

or 10%. Higher numbers indicates that the

comparisons are less consistent, while smaller

numbers mean comparison are more consistent, CR

above 0.1 or 10% indicates that the pairwise

comparisons should be revisited or reversed (Setiawan

et al., 2014).

Table 4: Pairwise Comparison Matrix for the Criteria and Consistency Ratio

RD BL RAM INS C nth root of

product

values

Priority CI RI CR

RD 1 1/3 1/9 1/5 1/4 0.283 0.05 0.08 1.12 0.07

BL 3 1 1 1 1 1.246 0.20

RAM 9 1 1 3 1 1.933 0.33

INS 5 1 1/3 1 2 1.270 0.22

Cost (C) 4 1 1 1/2 1 1.149 0.20

Totals 5.880 1.00

RD = Resolution Display (RD; BL = Battery Life (BL); RAM = Random Access Memory (RAM); INS = Internal Storage (INS)

(4) Ranking method: This is one of the simplest

approaches to assign criteria weights. The criteria are

usually ranked from best to worst importance. There

are three approaches to calculate weights using the

ranking method. They include rank sum, the rank

exponent and rank reciprocal. In the rank sum, the

weights are computed from the individual ranks

normalized by dividing the sum of the ranks. The

formula for rank sum determination can be expressed

as (Raszkowska, 2013):

n

k

k

j

j

pn

pn

RSw

1

1

1

)(

(6)

Where

j

p

is the rank of the j-th criterion, j = 1, 2, , n

The rank exponent weight (RE) method is a similar to

the rank sum method except that the value is raised to

an exponential of a parameter p which may be

estimated by a decision maker as a result of the most

important criterion. The formula for the rank exponent

method is given as (Raszkowska, 2013):

n

k

p

k

p

j

j

pn

pn

REw

1

)1(

)1(

)(

(7)

Where

j

p

is the rank of the j-th criterion, and p is the

parameter describing the weights,

j = 1, 2, …, n

The reciprocal (or inverse) weights (RR) method uses

the normalized reciprocal of the criterion rank. This

can be expressed as (Raszkowska, 2013):

n

k

k

j

j

p

p

RRw

1

)/1(

/1

)(

(8)

Where

j

p

is the rank of the j-th criterion, j = 1, 2, , n

For clearer understanding, an example on the

computation of weights using rank sum, rank exponent

and rank reciprocal method see Table 5. These

methods are actually not appropriate for a large

number of criteria due to the difficulty of straight

ranking. However, these techniques should be

regarded as weight approximation method only

because of its simplicity and gives an easy check of

criteria weights.

Table 5: Ranking methods

Rank Sum Rank exponent Rank reciprocal

1

j

pn

p

j

pn )1(

p= 2

j

p/1

Criteria Str.

Rank

Weight Normalized weight normalized weight normalized

Physics 5 2 0.065 4 0.044 0.20 0.082

Chemistry 2 5 0.161 25 0.275 0.50 0.204

Biology 4 3 0.097 9 0.099 0.25 0.102

Mathematics 6 1 0.032 1 0.011 0.17 0.068

English 3 4 0.129 16 0.176 0.33 0.136

Agric. Sc. 1 6 0.194 36 0.396 1.00 0.408

Total 31 91 2.45

Note: 1 is the most important criterion, and 6- is the least important out of the six criteria using the straight rank first

Weighting Methods for Multi-Criteria….. 1454

ODU, GO

(5) Ratio weighting method: The ratio method is one

of the subjective weighting methods that requires the

input of the decision makers to rank the relevant

criteria according to their importance. Here, the least

important criterion is assigned the value of 10, and the

other criteria are assigned multiples of 10. The

resulting weights are then normalized to sum to one.

(6) Swing weighting method: In the swing weighting

method, the decision maker is asked to select an

alternative with the worst outcome and picks the

criteria whose performance is likely to change (or

swing) from its worst to the best. The criterion with

the most preferred swing is given higher weight, e.g.,

100 points value. Next, the criteria whose performance

the decision maker would like to change from its worst

to the best level is selected again and a value between

0 and 100 representing its relative importance

regarding the most important criteria is provided. Then

obtain the average normalized weights and normalized

weights interval (Parnell and Trainor, 2009). A typical

template for swing weight matrix is given in Table 6.

The first step is to create a matrix in which the top

defines the values in terms of the relative importance

while the left hand side represents the range of

variation values. Assign a value measure that is most

preferred to the decision and at the same time has a

large variation to the upper left of the matrix labeled

cell ‘R’. And a value measure that has the worst

preferred importance and has the smallest variation in

its scale is placed to the lower right of the matrix

labeled cell ‘Z’.

Table 6: Swing weight matrix template

Importance of the value

measure to the decision makers

High Medium Low

Variation

of Scale

High R S T

Medium U V W

Low X Y Z

However, the consistency rule for swing weight

method is very important and necessary to ensure

consistency of the weights assigned to individual cells

of the matrix. And for the consistency rules to hold,

the following conditions or relationship of non-

normalized swing weights must be followed:

(a)

iR

CC

for all value of i in all other cell

(b)

ZWYVXU

CCCCCC ,,,

,

(c)

ZWYTVS

CCCCCC ,,,

,

(d)

ZYX

CCC ,

(e)

ZWYV

CCCC ,,

(f)

ZWT

CCC ,

(g)

ZY

CC

(h)

ZW

CC

In assigning swing weights, the stakeholders need to

make compromise between level of importance and

level of variation in measure scale. This is done by

allowing the stakeholders to assign arbitrary large

weight to the top left hand side of the matrix, for

example, 1000 or 100 as shown in Table 7 (

100

R

C

) and the weight of the lowest importance

in cell Z (

1

Z

C

). Then using the expression in

equation (9) to calculate the normalized swing value

for i

th

value.

n

i

i

i

i

C

C

W

1

(9)

Where

i

C

is the unnormalized swing weight.

Table 7: Element of the swing weight matrix

Importance of the value

measure to the decision makers

High Medium Low

Variation

of

Scale

High 100 85 35

Medium 80 40 20

Low 50 10 1

(7) Nominal Group Technique (NGT): Nominal group

technique is a structured brainstorming technique that

is used to produce a large number of ideas concerning

an issue and making sure that all the group members

have equal participation. Apart from the fact that the

technique can be used to generate a large number of

ideas, but can also be used to prioritize the ideas and

more importantly, the ideas which receive majority of

the votes are selected (Abdullah and Islam 2011). In

NGT the weights are derived by carrying out the

following steps with experience people in a group of

not less than seven members.

Step 1: Silent generation of ideas in writing: All

participants are given about 10 minutes to generate as

many ideas as possible with respect to the issue at hand

in absolute silence and done independently.

Step 2: Round-robin recording of ideas on a flip pad:

In this step, each participant is asked to provide the

best idea from the list generated in the first round. All

ideas are written down on a flip pad or marker board,

and this is carried out in subsequent in a round-robin

manner until all the ideas are exhausted in the

participants’ list.

Weighting Methods for Multi-Criteria….. 1455

ODU, GO

Step 3: Discussion of ideas for clarification: At this

step, all the ideas are being discussed for clarification.

The facilitator starts from the beginning of the master

list and asks the participants whether the meaning is

clear to them or not. And for whatever reason any of

the idea that is not clear, then it needs to be clarified

by the person who provided it or by someone else.

Step 4: Voting to select the most important ideas: This

last aspect of the technique is very important because

all the ideas selected will be used for determining the

criteria weight. The participant will be asked to select

5 most important ideas from the master list and rate

them using 1 to 5 scales according to their importance.

The most important idea is assigned a rating of 5 and

the least receive the rating of 1. This rating will be

done by each participant and then the aggregate voting

will be computed accordingly.

(8) Simple Multi-attribute Rating Technique

(SMART): The SMART technique is a compensatory

method of multiple criteria decision making originally

developed by Edward in 1971 (Patel et al. 2017). In

the SMART method, it is described as a process of

rating of alternatives and weighting criteria. So, we

will be looking at the weighting approach of this

method. In this method, decision maker is asked to

rank the criteria in terms of their importance from

worst to best. The least important a criterion is

assigned 10 points while the most important criterion

is given 100 points with an increasing number of

points are assigned to the other criteria according to

their importance. The criteria weight can be calculated

by normalizing the sum of the points to one.

Objective weighting methods: As mentioned earlier,

the objective weighting methods are derived from

information gathered from each criterion using a

mathematical function to compute the weights without

the interference of the decision maker. This includes

entropy method, mean weight, standard deviation,

statistical variance procedure, and criteria importance

through inter-criteria (CRITIC).

:

Entropy Method: The entropy method is the method

used for assessing the weight in a given problem

because with this method, the decision matrix for a set

of candidate materials contains a certain amount of

information. The entropy works based on a predefined

decision matrix. Entropy in information theory is a

criterion for the amount of uncertainty represented by

a discrete probability distribution, in which there is

agreement that a broad distribution represents more

uncertainty than does a sharply packed one (Deng et

al. 2000). The entropy method for assessing the

relative importance of criteria is calculated using

material data for each criterion, the entropy of the set

of normalized outcomes of the jth criterion is given by

)ln(/)ln(

1

mppE

m

i

ijijj

; j = 1, 2,

. . . ,n and i = 1, 2, . . . , m (10)

The

ij

p

form the normalized decision matrix and is

given by

m

i

ij

ij

ij

r

r

p

1

; i = 1, 2, . . . , and; j = 1,

2, . . . ,n (11)

Where

ij

r

is an element of the decision matrix, k is a

constant of the entropy equation and

j

E

as the

information entropy value for jth criteria. Hence, the

criteria weights,

j

w

is obtained using the following

expression.

n

ij

j

j

j

E

E

w

)1(

1

; j = 1,2,..,n (12)

Where (

j

E1

) is the degree of diversity of the

information involved in the outcomes of the jth

criterion.

(1) Mean Weight (MW): The mean weight (equal

importance) is mostly adopted when there is no

information from decision maker or when there is no

enough information available to reach a decision

(Jahan et al., 2012). The mean weight is based on the

assumption that all criteria are of equal importance.

This can be derived by using equation (13).

n

w

j

1

(13)

Where n is the number of criteria

(2) Standard Deviation Method: The standard

deviation method determines the weights of the

criteria in terms of their standard deviations using the

expression in equation (14) and (15) (Jahan et al.,

2012).

m

rr

m

i

jij

j

2

1

][

mi ,...,1

nj ,...,1

(14)

Therefore,

Weighting Methods for Multi-Criteria….. 1456

ODU, GO

n

j

j

j

j

w

1

(15)

where

j

is the standard deviation for criterion j

4. Statistical Variance Procedure: This is another

method of an objective weighting approach based on

statistical variance of information as given in equation

(16)

n

i

ijijj

xx

n

1

2

)(

1

(16)

Where

j

is the statistical variance;

ij

x

is the average

value of set of data

Hence, the weight of the criteria can be obtained as in

equation (17)

n

i

j

j

j

w

1

(17)

(3) Criteria importance through inter-criteria

(CRITIC): The criteria importance through inter-

criteria correlation (CRITIC) method is based on the

standard deviation proposed by Diakoulaki et al.

(1995) which uses correlation analysis to measure the

value of each criterion. First of all, normalized the

decision matrix using equation (18) and (19).

minmax

min

jj

jij

ij

yy

yy

mi ,...,1

;

nj ,...,1

For benefit criteria (18)

minmax

max

jj

ijj

ij

yy

yy

mi ,...,1

;

nj ,...,1

For cost criteria (19)

Then to calculate the weight of the criteria in equation

(21), we have to compute a linear correlation

coefficient between the criteria values in the matrix by

employing equation (20)

2

1

2

1

)()(

))((

m

i

kik

m

i

jij

kik

m

i

jij

jk

;

nkj ,...,1,

(20)

n

k

k

j

j

w

1

(21)

Where

n

k

jkjj

1

)1(

;

nj ,...,1

Integrated weighting method: The integrated or

combines weighting methods are derived from both

subjective and objective information on criteria

weights. The approach determines weights by solving

a mathematical model and takes into consideration

both subjective and objective factors. It overcomes the

shortages which occur in either a subjective or an

objective approach. Most times, the weights

determined by subjective method are bias and neglects

the objective information aspect. The judgment of the

decision makers sometimes depends on the knowledge

or experience and this may affect the decision process

to some extent. Therefore, literature has it that none of

the two approaches are perfect, and the integrated

method might be the most appropriate for determining

the criteria weights. There are some numbers of

combinations that have been proposed and developed

by scholars. Jian et al. (1999) proposed a subjective

and objective integrated approach which gives ranking

of alternatives that reflects both subjective

considerations and objective situations. Also, Jahan et

al. (2012) proposed an integrated weighting procedure

which was able to strengthen the existing MCDM

material selection especially when there are numerous

alternatives with inter-related criteria.

Conclusion: The paper mainly aims at the various

methods for determining criteria weights in MCDM by

considering subjective weighting method, objective

weighting method and integrated weights approach.

which allows the decision maker to assess the actual

performance of a particular selection process, and

makes it easier to identify the difference between the

subjective and objective weighting method, and the

expected level of performance that intend to achieve

in the future.

REFERENCES

Abdullah, M.M.B; Islam, R. (2011). Nominal Group

Technique and its Applications in Managing Quality in

Higher Education. Pak. J. Com. Soc. Sci. 5(1), 81-99.

Aldian, A; Taylor, M.A.P. (2005). A consistent method to

determine flexible criteria weights for multicriteria

transport project evaluation in developing countries. J.

East. Asia Soc. Transport. Stud. 6: 3948-3963.

Weighting Methods for Multi-Criteria….. 1457

ODU, GO

Arbel, A. (1989). Approximate articulation of preference

and priority derivation. European J. Operate. Res. 43,

317–326.

Arvind, J; Janpriy, S, (2018). A Comprehensive literature

review of MCDM techniques ELECTRE,

PROMETHEE, VIKOR, and TOPSIS applications in

business competitive environment. Inter. J. Cur. Res.

10(2) 65461-65477

Chatterjee, P; Chakraborty, S. (2012). Material selection

using preferential ranking methods. Mat. Design 35:

384-393.

Davood, S; John, E; Rajkumar, R (2005). A review of multi-

criteria making methods for enhanced maintenance

delivery. Procedia CIRP 37, 30-35.

Diakoulaki, D; Mavrotas, G; Papayannakis, L. (1995).

Determining Objective Weights in Multiple Criteria

Problems: The Critic Method. Computer & Operation

Research 22:763–770.

Diakoulaki, D; Mavrotas, G; Lefteris, P. (1995).

Determining objective weights in multiple criteria

problems: The CRITIC method. Comp. Operate. Res.

22 (7) 763-770.

Dong, S.J; Jine, S.J; Eui, H.K. (2005). Development of

Integrated Materials Database System for Plant

Facilities Maintenance and Optimisation. Key

Engineering Materials, Switzerland: Trans Technical

Publications 297-300: 2681-2686.

Dragan, P; Zeljko, S; Sinisa, S. (2018). A new model for

determining weight coefficient of criteria in MCDM

models: Full consistency method (FUCOM). Symmetry

10(9), 393

Ginevicius, R; Podvezko, V. (2005). Objective and

subjective approaches to determining the criterion

weight in multicriteria models. Proceedings of

International Conference RelStat. Transport and

Telecommunication 6(1) 133-137.

Jahanshahloo, G.R; Lotfi, F. H; Izadikhah, M. (2006). An

algorithmic method to extend TOPSIS for decision-

making problems with interval data, Applied

Mathematics and Computation 175.2: 1375-1384

Jian, M; Zhi-Ping, F; Li-Hua, H. (1999). A subjective and

objective integrated approach to determine attribute

weights. European Journal of Operational Research,

112(2), 397-404.

Liu, W; Zeng, L (2008). A new TOPSIS method for fuzzy

multiple attribute group decision making problem, J.

Guilin Univ. Electro. Technol. 28.1: 59-62.

Mirko, S; Edmundas, K.Z; Dragan, P; Zeljko, S; Abbas, M.

(2019). Application of MCDM methods in

sustainability engineering: A literature review,

Symmetry 2008-2018.

Olson, L.D. (2008). Subjectivity in Multiple Criteria

Decision Analysis. Human Centered Processes.

Parnell, G. and Tainor, T. (2009). Using the Swing Weight

to Weight Multiple Objective. Proceeding of the

INCOSE international Symposium, Singapore.

Patel, R.M; Vashi, P.M; Bhatt, V.B. (2017). SMART- Multi-

criteria decision making technique for use in planning

activities. NHCE.

Rao, R.V; Davim, J.P. (2008). A decision making

framework model for material selection using a

combined multiple attribute decision making method.

Intern. J. Adv. Technol. 35: 751-760.

Rao, R.V; Patel, B.K. (2011). Material Selection Using a

Novel Multiple Attribute Decision Making Method.

Inter. J. Man. Mat. Mech. Engineer. 1.1: 43-56.

Roszkowska, W. (2013). Rank Ordering Criteria Weighting

Methods-A Comparative Overview. Optimum Studia

Ekonomiczne NR, 5(65).

Saaty, T.L. (1977). A scaling method for priorities in

heirachical structures. J. Math. Psychology 15(3) 234-

281.

Saaty, T.L. (1980). The Analytic Hierarchy Process:

Planning, Priority Setting, Resource Allocation,

McGraw-Hill.

Setiawan, A; Sediyono, E; Moekoe, D.A.L (2014).

Application of AHP method in determining priorities of

conversion of unused land to food land in Minahasa

Tenggara. International Journal of Computer

Applications 89(8), 37-44.

Starr, M.K, Greenwood, L.H (1977). Normative generation

of alternatives with multiple criteria evaluation.

Multiple criteria decision making. Starr M.K, Zeleny

M, Eds.. New York, North-Holland. 111–112.

Vinodh, S; Prasanna, M; Praskash, N. H. (2014). Integrated

Fuzzy AHP-TOPSIS for selecting the best plastic

recycling methods. A case study. Applied Mathematics

Modeling, 38(19, 20):4662-4672.

Zardari, N.H; Ahmed, K; Shirazi, S.M; Yusop, Z.B. (2005).

Weighting methods and their effects on multi-criteria

decision making models outcomes in water resources

management. Springer Briefs in Water Sciences and

Technology.

Xu, Z. (2007). Methods for aggregating interval-valued

intuitionistic fuzzy information and their application to

decision making. Control and Decision 22.2: 215-219.