Applied mathematics / English

Entropy Method (for weight determination of criteria/Attributes )

Manoj Mathew · YouTube · 4:36

YouTube · Manoj MathewOriginal video

IN THIS VIDEO

01Entropy method for objective weights
02Step 1: Normalize the decision matrix
03Step 2: Entropy formula and scaling constant

FROM THE LEARNING NOTES

r_{ij}=\frac{x_{ij}}{\sum_{i=1}^{m}x_{ij}}

Step 1: Normalize the decision matrix

SELECTED MOMENT

Step 2: Compute entropy values

SUGGESTED PREPARATION

Basic matrices and summation notation
Natural logarithm $\ln$
Multi-criteria decision-making context
Familiarity with AHP is helpful for the contrast made in the introduction
Normalized decision matrix
Logarithmic functions
Basic summation notation

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Reviewed learning material · Video analysis · English

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This 180-second lecture segment introduces the entropy method for determining objective criterion weights in multi-criteria decision making, contrasts it with AHP, and works through a mobile-phone selection example up to normalization and entropy computation. The clip shows the raw $5\times 4$ decision matrix, derives the normalized matrix using column sums, then computes entropies $e_j$ for all four criteria. Final weight determination is not reached within this excerpt. This video segment demonstrates the final steps of the Entropy Method for determining objective weights of criteria. It begins by showing the calculated entropy values ( $e_j$ ) for four criteria: Price or Cost, Storage Space, Camera, and Looks. The speaker then introduces the concept of 'degree of diversification', defined as $1 - e_j$ , and calculates it for each criterion. Finally, the video applies the weight vector formula, $w_j = (1 - e_j) / \sum(1 - e_j)$ , to normalize the degrees of diversification, yielding the final objective weights for each criterion. The process is illustrated with a step-by-step numerical example using a table that updates with the calculated values.

Use the learning inspector for key ideas and moments, or open the reading tabs for the complete notes.

Chapters

0:00Introduction to entropy-based objective weights 1:03Example decision matrix for mobile phones 1:17Step 1: Normalize the decision matrix 1:44Step 2: Compute entropy values 2:36Column sums and final entropies 3:00Recap of Entropy Calculation 3:13Degree of Diversification 3:44Calculating Final Objective Weights 4:24Outro

Learning script

Generated from the video's visuals and explanation; not verbatim speech.

The segment opens by framing the entropy method as a tool for assigning criterion weights in multi-criteria decision making when decision makers disagree about importance. The speaker explicitly contrasts this with AHP, noting that AHP requires subjective pairwise comparison judgments, whereas the entropy method derives weights from the data pattern itself.

Entropy weight is described as measuring how close the alternatives are to one another with respect to a given criterion. The speaker adds that the idea comes from transportation models, where entropy measures dispersion of trips between origins and destinations, and concludes that the resulting weights are called objective weights.

A concrete example appears: five mobile phones are compared across four criteria—Price or Cost, Storage Space, Camera, and Looks. The raw matrix is shown first, establishing the input data for the method.

Step 1 applies column-wise normalization using $r_{ij}=\frac{x_{ij}}{\sum_{i=1}^{m}x_{ij}}$ . The video computes the column sums $1250$ , $112$ , $60$ , and $18$ , then rewrites each entry as a fraction over its column sum and finally as a decimal. This produces a normalized matrix whose columns sum to 1.

Step 2 introduces the entropy formula $e_j=-h\sum_{i=1}^{m}r_{ij}\ln r_{ij}$ with $j=1,2,\ldots,n$ , together with the scaling constant $h=\frac{1}{\ln(m)}=\frac{1}{\ln(5)}=0.621335$ . The speaker explains the procedure verbally while the slide shows each normalized value multiplied by its natural logarithm.

The computation proceeds cell by cell. For example, in the Price or Cost column, terms such as $0.2000\ln(0.2000)$ , $0.1600\ln(0.1600)$ , and so on are evaluated, yielding negative products because $0<r_{ij}<1$ .

The column sums of $r_{ij}\ln r_{ij}$ are then displayed as $-1.59938$ , $-1.54983$ , $-1.56413$ , and $-1.56706$ . Multiplying each by $-h$ gives the entropies $e_1=0.99375$ , $e_2=0.96296$ , $e_3=0.97185$ , and $e_4=0.97367$ .

The clip ends immediately after the entropy row is shown. It does not continue to redundancy values, final weights, or ranking of the mobile phones.

The video starts by reviewing the entropy calculation. The speaker points out the constant $h = -0.621335$ and explains that multiplying it by the sum of $r_{ij} \ln r_{ij}$ yields the entropy values $e_j$ for each criterion, which are displayed in the table.

Next, the concept of 'degree of diversification' is introduced. The speaker explains that it is calculated as $1 - e_j$ . The table updates to show these new values for each criterion: Price or Cost (0.00625), Storage Space (0.03704), Camera (0.02815), and Looks (0.02633).

To find the final objective weights ( $w_j$ ), the speaker sums all the degrees of diversification, resulting in $0.09777$ . Then, each individual degree of diversification is divided by this total sum. The formula $w_j = \frac{1 - e_j}{\sum_{j=1}^n (1 - e_j)}$ is shown on screen.

The calculated weights appear in the final row of the table. The speaker reads them out as percentages: Price or Cost gets about 6.3%, Storage Space gets 37.88%, Camera gets 28.7%, and Looks gets 26.9%. These values represent the objective importance of each criterion based on the data's variability.

The video concludes with a standard outro asking viewers to subscribe and enable notifications.

Knowledge cards

Entropy method for objective weights

The entropy method is presented as a data-driven way to determine criterion weights in multi-criteria decision making. Unlike AHP, it does not require pairwise comparison opinions from the decision maker. Entropy weight measures how similar the alternatives are with respect to a criterion, and the resulting weights are called objective weights.

Step 1: Normalize the decision matrix

Before computing entropy, each column of the raw decision matrix is normalized by dividing every entry by the sum of that column. In the mobile-phone example, the column sums are 1250, 112, 60, and 18, and the normalized matrix has columns that sum to 1.

r_{ij}=\frac{x_{ij}}{\sum_{i=1}^{m}x_{ij}}

Step 2: Entropy formula and scaling constant

For each criterion j, entropy is computed from the normalized values by summing r_{ij} $\ln r$ _{ij} over all alternatives and multiplying by -h. The constant h is defined using the number of alternatives m; here $m=5$ , so $h=1/\ln(5)=0.621335$ .

e_j=-h\sum_{i=1}^{m}r_{ij}\ln r_{ij},\quad h=\frac{1}{\ln(m)}

Worked entropy results for the example

Applying the formula to the normalized mobile-phone matrix gives column sums -1.59938, -1.54983, -1.56413, and -1.56706. After multiplication by -h, the entropies are 0.99375 for Price or Cost, 0.96296 for Storage Space, 0.97185 for Camera, and 0.97367 for Looks.

(e_1,e_2,e_3,e_4)=(0.99375,0.96296,0.97185,0.97367)

Entropy Formula

The entropy $e_j$ for a criterion is calculated using the formula $e_j = -h \sum_{i=1}^m r_{ij} \ln r_{ij}$ , where $h$ is a constant ( $1/\ln(m)$ ) and $r_{ij}$ are the normalized values of the alternatives for that criterion. This step quantifies the amount of information or uncertainty associated with each criterion.

e_j = -h \sum_{i=1}^m r_{ij} \ln r_{ij}

Degree of Diversification

The degree of diversification, denoted as $d_j = 1 - e_j$ , measures the usefulness of a criterion. A lower entropy (more information) leads to a higher degree of diversification, indicating that the criterion provides more distinguishing power among the alternatives.

d_j = 1 - e_j

Objective Weight Calculation

The final objective weight $w_j$ for each criterion is found by normalizing its degree of diversification. The formula is $w_j = \frac{1 - e_j}{\sum_{j=1}^n (1 - e_j)}$ . This ensures that the sum of all weights equals 1, representing the relative importance of each criterion in the decision-making process.

w_j = \frac{1 - e_j}{\sum_{j=1}^n (1 - e_j)}

Detailed learning notes

Explore conditions, steps and evidence. Supplementary explanations are labeled separately from content shown in the video.

Symbols · 11

x_{ij}

Clear evidence

Shown in the video

Evidence

Formula
Observation
Normalization formula shown as r_{ij}=x_{ij}/\sum_{ $i=1$ }^{m}x_{ij}.
Audio
Observation
The speaker says the denominator is the sum of x i j divided in each column.

Symbol

x_{ij}

Meaning

Original decision-matrix entry for alternative i and criterion j.

Domain

Nonnegative numerical attribute values; in this example includes price, storage, camera, and looks ratings.

r_{ij}

Clear evidence

Shown in the video

Evidence

Formula
Observation
Formula defines r_{ij}.
Diagram
Observation
Normalized table entries are displayed after division by column sums.

Symbol

r_{ij}

Meaning

Normalized value of x_{ij} within its column.

Domain

Values between 0 and 1; each criterion column sums to 1.

h

Clear evidence

Shown in the video

Evidence

Formula
Observation
Entropy formula contains factor -h.
Formula
Observation
Auxiliary formula shows $h=1/\ln(m)=1/\ln(5)=0.621335$ .
Audio
Observation
Speaker states h is one upon log m, where m is the number of alternatives.

Symbol

Meaning

Scaling constant used in entropy computation.

Domain

Positive real constant; here $h=0.621335$ for $m=5$ .

m

Clear evidence

Shown in the video

Evidence

Formula
Observation
$h=1/\ln(m)$ with note that m is number of alternatives.
Audio
Observation
Speaker says in this example there are five alternatives, so $m=5$ .

Symbol

Meaning

Number of alternatives (rows) in the decision matrix.

Domain

Positive integer; here $m=5$ .

n

Clear evidence

Shown in the video

Evidence

Formula
Observation
Entropy formula states $j=1$ ,2, $\ldots$ ,n.

Uncertainties

The video does not explicitly define n in audio; it is inferred from the index range in the displayed formula.

Symbol

Meaning

Number of criteria or attributes (columns).

Domain

Positive integer; here $n=4$ .

$e_j$

Clear evidence

Shown in the video

Evidence

Formula
Observation
Entropy formula is $e_j=-h\sum_{i=1}^{m}r$ _{ij} $\ln r$ _{ij}.
Diagram
Observation
Row labeled $e_j$ displays entropy values for each criterion.

Symbol

$e_j$

Meaning

Entropy value for criterion j.

Domain

Real number in [0,1] under the given normalization and scaling.

$\sum_{i=1}^{m} r$ _{ij} $\ln r$ _{ij}

Clear evidence

Shown in the video

Evidence

Formula
Observation
Column-sum row is labeled $\sum_{i=1}^{m} r$ _{ij} $\ln r$ _{ij}.
Diagram
Observation
Negative column sums are shown before multiplication by -h.

Symbol

$\sum_{i=1}^{m} r$ _{ij} $\ln r$ _{ij}

Meaning

Column-wise sum of products r_{ij} $\ln r$ _{ij} for criterion j.

Domain

Negative real values in this example because $0<r$ _{ij}<1.

$e_j$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The symbol $e_j$ is shown in the table and in the formula for entropy.
Audio
Observation
The speaker refers to 'the $e_j$ value, that is the entropy value'.

Symbol

$e_j$

Meaning

Entropy value for the j-th criterion

Domain

Positive real number

h

Clear evidence

Shown in the video

Evidence

Formula
Observation
The symbol h is used in the entropy formula $e_j = -h \sum r$ _{ij} $\ln r$ _{ij}.
Audio
Observation
The speaker says 'that is minus h which is minus point six two one three three'.

Symbol

Meaning

Constant multiplier in the entropy formula

Domain

Positive real number

$d_j$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The symbol $d_j$ is shown in the table as $d_j = 1 - e_j$ .
Audio
Observation
The speaker states 'Here 1 minus $e_j$ is known as degree of diversification'.

Symbol

$d_j$

Meaning

Degree of diversification for the j-th criterion

Domain

Positive real number

$w_j$

Clear evidence

Shown in the video

Evidence

Formula
Observation
The symbol $w_j$ is shown in the table and in the formula for the weight vector.
Audio
Observation
The speaker mentions 'to calculate the final weights, that is $w_j$ '.

Symbol

$w_j$

Meaning

Objective weight for the j-th criterion

Domain

Positive real number

Knowledge points · 6

Entropy method for objective weight determination

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker introduces a method to calculate weights of criteria when decision makers have conflicting views on the value of weights.
Audio
Observation
Speaker contrasts this with AHP, saying pairwise comparison opinions are needed there but not in entropy method.
Audio
Observation
Speaker says entropy weight describes how much different alternatives approach one another with respect to a certain criteria.
Audio
Observation
Speaker says the weights calculated by this method are also known as objective weights.

Definition

Explanation

The video presents the entropy method as a way to determine criterion weights without eliciting pairwise judgments from decision makers. It defines entropy weight as a parameter describing how close alternatives are to one another with respect to a given criterion, and calls the resulting weights objective weights.

Formula

Conditions

Used in multi-criteria decision making.
Applies when criteria weights must be derived from data rather than subjective pairwise comparisons.

Step 1: Normalize the decision matrix

Clear evidence

Shown in the video

Evidence

Formula
Observation
Displayed formula r_{ij}=x_{ij}/\sum_{ $i=1$ }^{m}x_{ij}.
Audio
Observation
Speaker says first calculate the sum value in the column, then divide each value by that sum.
Diagram
Observation
Column sums 1250, 112, 60, 18 appear beneath the original matrix.

Formula

Explanation

Each entry is divided by the sum of its own column, producing a normalized matrix whose columns sum to 1. This prepares the raw decision matrix for entropy computation.

Formula

r_{ij}=\frac{x_{ij}}{\sum_{i=1}^{m}x_{ij}}

Conditions

Apply column-wise normalization.
Each column sum must be nonzero.
In this example $m=5$ alternatives and $n=4$ criteria.

Prerequisites

Entropy method for objective weight determination

Step 2: Compute entropy for each criterion

Clear evidence

Shown in the video

Evidence

Formula
Observation
Displayed formula $e_j=-h\sum_{i=1}^{m}r$ _{ij} $\ln r$ _{ij}, $\quad j=1$ ,2, $\ldots$ ,n.
Formula
Observation
Auxiliary formula $h=1/\ln(m)=1/\ln(5)=0.621335$ .
Audio
Observation
Speaker explains multiplying each cell value by the log of that same value, summing each column, then multiplying by negative h.

Formula

Explanation

For each criterion column, compute the sum of r_{ij} $\ln r$ _{ij} over all alternatives, then multiply by -h to obtain the entropy $e_j$ . The constant h normalizes the entropy using the number of alternatives.

Formula

e_j=-h\sum_{i=1}^{m}r_{ij}\ln r_{ij},\quad j=1,2,\ldots,n

Conditions

Use the normalized matrix entries r_{ij}.
Use natural logarithm $\ln$ as shown in the displayed formulas.
$h=1/\ln(m)$ ; here $m=5$ so $h=0.621335$ .

Prerequisites

Step 1: Normalize the decision matrix

Entropy Formula

Clear evidence

Shown in the video

Evidence

Formula
Observation
The formula $e_j = -h \sum_{i=1}^m r$ _{ij} $\ln r$ _{ij}, $j = 1$ , 2, ..., n is displayed on the screen.
Audio
Observation
The speaker explains how to compute the entropy values using the given formula and constant h.

Formula

Explanation

The entropy for each criterion is calculated using the formula $e_j = -h \sum_{i=1}^m r$ _{ij} $\ln r$ _{ij}. The constant h is given as $1 / \ln(m)$ , where m is the number of alternatives.

Formula

e_j = -h \sum_{i=1}^m r_{ij} \ln r_{ij}

Conditions

m is the number of alternatives
n is the number of criteria
r_{ij} is the normalized value of the i-th alternative for the j-th criterion

Degree of Diversification

Clear evidence

Shown in the video

Evidence

Formula
Observation
The formula $d_j = 1 - e_j$ is displayed in the table.
Audio
Observation
The speaker explicitly defines '1 minus $e_j$ ' as the 'degree of diversification'.

Definition

Explanation

The degree of diversification for a criterion is defined as 1 minus its entropy value. It represents the amount of useful information provided by the criterion.

Formula

d_j = 1 - e_j

Conditions

$e_j$ is the entropy value of the j-th criterion

Prerequisites

Entropy Formula

Weight Vector Formula

Clear evidence

Shown in the video

Evidence

Formula
Observation
The formula $w_j = \frac{1 - e_j}{\sum_{j=1}^n (1 - e_j)}$ is displayed on the screen.
Audio
Observation
The speaker explains that the final weights are calculated by dividing the degree of diversification of each criterion by the sum of all degrees of diversification.

Formula

Explanation

The objective weight for each criterion is calculated by normalizing its degree of diversification. The formula is $w_j = \frac{1 - e_j}{\sum_{j=1}^n (1 - e_j)}$ .

Formula

w_j = \frac{1 - e_j}{\sum_{j=1}^n (1 - e_j)}

Conditions

n is the total number of criteria
$e_j$ is the entropy value of the j-th criterion

Prerequisites

Degree of Diversification

Claims and conditions · 2

Entropy-derived weights are called objective weights

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker states: The weights calculated by this method are also known as objective weights.

Proposition

Statement

Weights obtained by the entropy method are referred to as objective weights.

Hypotheses

The entropy method is applied to a decision matrix.

Quantifiers

For the method described in the video.

Entropy method avoids explicit pairwise opinion elicitation

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker says: In entropy method, no such option is needed, referring to giving opinions for pairwise comparison matrices in AHP.

Proposition

Statement

Unlike AHP, the entropy method does not require the decision maker to provide pairwise comparison opinions.

Hypotheses

Comparison is between the entropy method and the AHP process as described in the video.

Quantifiers

For the methods discussed in the introduction.

Derivations and proofs · 4

Derivation of the normalized matrix from the example data

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Original matrix values and column sums are shown.
Diagram
Observation
Intermediate fractions such as $250/1250$ , $16/112$ , $12/60$ , $5/18$ appear.
Diagram
Observation
Final normalized matrix values are displayed.

Numerical verification

Steps

Expression
$\sum_{i=1}^{5}x_{i1}=250+200+300+275+225=1250$
Explanation
Sum the Price or Cost column.
Justification
Column sum required by the normalization formula.
Shown in the video
Expression
$\sum_{i=1}^{5}x_{i2}=16+16+32+32+16=112$
Explanation
Sum the Storage Space column.
Justification
Column sum required by the normalization formula.
Shown in the video
Expression
$\sum_{i=1}^{5}x_{i3}=12+8+16+8+16=60$
Explanation
Sum the Camera column.
Justification
Column sum required by the normalization formula.
Shown in the video
Expression
$\sum_{i=1}^{5}x_{i4}=5+3+4+4+2=18$
Explanation
Sum the Looks column.
Justification
Column sum required by the normalization formula.
Shown in the video
Expression
$r_{11}=\frac{250}{1250}=0.2000,\; r_{12}=\frac{16}{112}=0.1429,\; r_{13}=\frac{12}{60}=0.2000,\; r_{14}=\frac{5}{18}=0.2778$
Explanation
Show the first row of normalized values as examples of the division step.
Justification
Direct application of r_{ij}=x_{ij}/\sum_i x_{ij}.
Shown in the video

Conclusion

The original $5\times 4$ decision matrix is transformed into a normalized matrix with column sums equal to 1.

Entropy computation for the Price or Cost column

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Products r_{ij} $\ln r$ _{ij} are displayed cell by cell.
Diagram
Observation
Column sum row shows -1.59938 for Price or Cost.
Diagram
Observation
Entropy row shows $e_1=0.99375$ .
Formula
Observation
$e_j=-h\sum_i r$ _{ij} $\ln r$ _{ij} and $h=0.621335$ .

Numerical verification

Steps

Expression
$0.2000\ln(0.2000)=-0.32189$
Explanation
Compute the first product in the Price or Cost column.
Justification
Term-by-term evaluation of r_{ij} $\ln r$ _{ij}.
Shown in the video
Expression
$0.1600\ln(0.1600)=-0.29321$
Explanation
Compute the second product in the same column.
Justification
Term-by-term evaluation of r_{ij} $\ln r$ _{ij}.
Shown in the video
Expression
$0.2400\ln(0.2400)=-0.34251$
Explanation
Compute the third product.
Justification
Term-by-term evaluation of r_{ij} $\ln r$ _{ij}.
Shown in the video
Expression
$0.2200\ln(0.2200)=-0.33311$
Explanation
Compute the fourth product.
Justification
Term-by-term evaluation of r_{ij} $\ln r$ _{ij}.
Shown in the video
Expression
$0.1800\ln(0.1800)=-0.30866$
Explanation
Compute the fifth product.
Justification
Term-by-term evaluation of r_{ij} $\ln r$ _{ij}.
Shown in the video
Expression
$\sum_{i=1}^{5}r_{i1}\ln r_{i1}=-1.59938$
Explanation
Add the five products to get the column sum.
Justification
Definition of the summation inside the entropy formula.
Shown in the video
Expression
$e_1=-h(-1.59938)=0.621335\times1.59938=0.99375$
Explanation
Multiply the negative column sum by h to obtain entropy for criterion 1.
Justification
Apply $e_j=-h\sum_i r$ _{ij} $\ln r$ _{ij} with $h=0.621335$ .
Shown in the video

Conclusion

The entropy of the Price or Cost criterion is $e_1=0.99375$ .

Entropy computation for the remaining criteria

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Column sums shown as -1.54983, -1.56413, -1.56706.
Diagram
Observation
Entropy row shows 0.96296, 0.97185, 0.97367.
Formula
Observation
Same entropy formula applies to every column j.

Numerical verification

Steps

Expression
$\sum_{i=1}^{5}r_{i2}\ln r_{i2}=-1.54983$
Explanation
Column sum for Storage Space is displayed.
Justification
Summation of the five products in that column.
Shown in the video
Expression
$e_2=-h(-1.54983)=0.96296$
Explanation
Entropy for Storage Space follows from multiplying by -h.
Justification
Apply the entropy formula with $h=0.621335$ .
Shown in the video
Expression
$\sum_{i=1}^{5}r_{i3}\ln r_{i3}=-1.56413$
Explanation
Column sum for Camera is displayed.
Justification
Summation of the five products in that column.
Shown in the video
Expression
$e_3=-h(-1.56413)=0.97185$
Explanation
Entropy for Camera follows from multiplying by -h.
Justification
Apply the entropy formula with $h=0.621335$ .
Shown in the video
Expression
$\sum_{i=1}^{5}r_{i4}\ln r_{i4}=-1.56706$
Explanation
Column sum for Looks is displayed.
Justification
Summation of the five products in that column.
Shown in the video
Expression
$e_4=-h(-1.56706)=0.97367$
Explanation
Entropy for Looks follows from multiplying by -h.
Justification
Apply the entropy formula with $h=0.621335$ .
Shown in the video

Conclusion

The entropies for the remaining criteria are $e_2=0.96296$ , $e_3=0.97185$ , and $e_4=0.97367$ .

Derivation of Objective Weights

Clear evidence

Shown in the video

Evidence

Formula
Observation
The table shows the step-by-step calculation: first computing $1 - e_j$ for each criterion, then summing these values to get 0.09777, and finally dividing each $1 - e_j$ by the sum to get $w_j$ .
Audio
Observation
The speaker narrates the process: 'On adding all these value, you will get this value that is point zero nine seven seven seven. I have divided point zero nine seven seven seven with all these degree of diversification values to calculate the final weights, that is $w_j$ .'

Numerical verification

Steps

Expression
$d_j = 1 - e_j$
Explanation
Calculate the degree of diversification for each criterion by subtracting its entropy from 1.
Justification
Definition of degree of diversification.
Shown in the video
Expression
$\sum_{j=1}^n d_j = 0.00625 + 0.03704 + 0.02815 + 0.02633 = 0.09777$
Explanation
Sum the degrees of diversification for all criteria.
Justification
Required for the denominator in the weight formula.
Shown in the video
Expression
$w_j = \frac{d_j}{\sum_{j=1}^n d_j}$
Explanation
Divide each criterion's degree of diversification by the total sum to find its objective weight.
Justification
Formula for calculating the weight vector.
Shown in the video

Conclusion

The objective weights for Price or Cost, Storage Space, Camera, and Looks are 0.0639154, 0.378831278, 0.287921, and 0.269333, respectively.

Worked examples · 2

Mobile-phone selection example up to entropy calculation

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Slide title reads Entropy Method and shows $a 5\times 4$ decision matrix for Mobile 1 to Mobile 5 across Price or Cost, Storage Space, Camera, Looks.
Audio
Observation
Speaker says: We want to select the best mobile phone out of all available five alternatives.
Diagram
Observation
Step 1 and Step 2 tables and formulas are shown sequentially.

Problem

Given a decision matrix for five mobile phones evaluated on four criteria, apply the entropy method through normalization and entropy computation.

Given

Alternatives: Mobile 1, Mobile 2, Mobile 3, Mobile 4, Mobile 5.
Criteria: Price or Cost, Storage Space, Camera, Looks.
Original matrix entries: [[250,16,12,5],[200,16,8,3],[300,32,16,4],[275,32,8,4],[225,16,16,2]] with units shown as $, GB, MP, and rating.
$m=5$ alternatives.
$n=4$ criteria.

Goal

Normalize the decision matrix and compute the entropy $e_j$ for each criterion.

Steps

Expression
$\sum x_{ij}=(1250,112,60,18)$
Explanation
Compute column sums of the original matrix.
Justification
Required denominators for normalization.
Shown in the video
Expression
$r_{ij}=\frac{x_{ij}}{\sum_{i=1}^{5}x_{ij}}$
Explanation
Divide each entry by its column sum.
Justification
Normalization formula shown on screen.
Shown in the video
Expression
$R=\begin{bmatrix}0.2000&0.1429&0.2000&0.2778\\0.1600&0.1429&0.1333&0.1667\\0.2400&0.2857&0.2667&0.2222\\0.2200&0.2857&0.1333&0.2222\\0.1800&0.1429&0.2667&0.1111\end{bmatrix}$
Explanation
Display the normalized matrix.
Justification
Result of applying the normalization formula to all cells.
Shown in the video
Expression
$h=\frac{1}{\ln(5)}=0.621335$
Explanation
Compute the entropy scaling constant.
Justification
Auxiliary formula shown on screen.
Shown in the video
Expression
$\sum_{i=1}^{5}r_{ij}\ln r_{ij}=(-1.59938,-1.54983,-1.56413,-1.56706)$
Explanation
Compute the column sums of r_{ij} $\ln r$ _{ij}.
Justification
Intermediate step in the entropy formula.
Shown in the video
Expression
$e_j=(-0.621335)(-1.59938,\,-1.54983,\,-1.56413,\,-1.56706)=(0.99375,0.96296,0.97185,0.97367)$
Explanation
Multiply each column sum by -h to obtain entropies.
Justification
Apply $e_j=-h\sum_i r$ _{ij} $\ln r$ _{ij}.
Shown in the video

Answer

Normalized matrix obtained; entropies are $e_1=0.99375$ , $e_2=0.96296$ , $e_3=0.97185$ , $e_4=0.97367$ .

Verification

Column sums of the normalized matrix are consistent with the displayed divisions, and each entropy equals -h times the displayed column sum.

Applying the Entropy Method to Determine Criteria Weights

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A table titled 'Entropy Method' shows four criteria: Price or Cost, Storage Space, Camera, and Looks. It displays intermediate calculations for $\sum r$ _{ij} $\ln r$ _{ij}, $e_j$ , $d_j = 1 - e_j$ , and the final weights $w_j$ .
Audio
Observation
The speaker walks through the calculation of entropy, degree of diversification, and final weights for the given criteria.

Problem

Determine the objective weights for four criteria (Price or Cost, Storage Space, Camera, Looks) using the Entropy Method.

Given

Entropy values ( $e_j$ ): 0.99375, 0.96296, 0.97185, 0.97367
Scaling constant $h = 0.621335 = 1/\ln (5)$ ; the entropy formula uses -h.

Goal

Calculate the final objective weights ( $w_j$ ) for each criterion.

Steps

Expression
$d_j = 1 - e_j$
Explanation
Compute the degree of diversification for each criterion.
Justification
Definition of degree of diversification.
Shown in the video
Expression
$\sum d_j = 0.09777$
Explanation
Sum the degrees of diversification.
Justification
Denominator for the weight calculation.
Shown in the video
Expression
$w_j = d_j / \sum d_j$
Explanation
Calculate the final weight for each criterion.
Justification
Weight vector formula.
Shown in the video

Answer

The final objective weights are: Price or Cost = 0.0639154, Storage Space = 0.378831278, Camera = 0.287921, Looks = 0.269333.

Verification

The sum of the calculated weights is approximately $1 (0.0639154 + 0.378831278 + 0.287921 + 0.269333 = 1.000000678)$ .

Visual events · 5

Title slide introducing the topic

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Static title slide with text MULTI CRITERIA DECISION MAKING and ENTROPY METHOD FOR WEIGHT DETERMINATION, plus author name Manoj Mathew.

Objects

Title text
Author name
Colored geometric background

Changes

No visual change during the introductory narration.

Invariants

Topic remains multi-criteria decision making and entropy-based weight determination.

Interpretation

The opening slide establishes the lecture subject before any numerical example appears.

Raw decision matrix displayed

Clear evidence

Shown in the video

Evidence

Diagram
Observation
A 5-row by 4-column table appears with headers Attribute Or Criteria, Price or Cost, Storage Space, Camera, Looks and rows Mobile 1 to Mobile 5.
Caption evidence
Observation
Bottom text reads Step1) Normalize the decision matrix.

Objects

Decision matrix table
Criterion headers
Alternative labels
Step 1 caption

Changes

The slide changes from title card to a tabular example.

Invariants

Five alternatives and four criteria remain fixed.

Interpretation

The example introduces the data structure on which the entropy method will operate.

Visual transformation from raw values to normalized values

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Column-sum row appears beneath the matrix.
Diagram
Observation
Cells temporarily show fractions like $250/1250$ and $16/112$ .
Diagram
Observation
Fractions are replaced by decimal normalized values.
Formula
Observation
Formula r_{ij}=x_{ij}/\sum_{ $i=1$ }^{m}x_{ij} is shown below the table.

Objects

Original matrix
Column sums
Fraction expressions
Decimal normalized matrix
Normalization formula

Changes

First column sums are added.
Then each cell is rewritten as a fraction over its column sum.
Finally fractions are replaced by decimal values.

Invariants

Matrix dimensions stay $5\times 4$ .
Column-wise normalization is applied throughout.

Interpretation

The animation makes explicit that normalization is performed independently in each criterion column.

Visual computation of entropy column by column

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Caption changes to Step 2) Compute entropy.
Formula
Observation
Entropy formula and h formula appear below the table.
Diagram
Observation
Cells show products such as 0.2000 * Ln(0.2000).
Diagram
Observation
Products are replaced by negative numeric values.
Diagram
Observation
A row of column sums appears, followed by a row labeled $e_j$ .

Objects

Normalized matrix
Entropy formula
h formula
Product expressions
Negative product values
Column-sum row
Entropy row $e_j$

Changes

Formula panel appears.
Each normalized entry is paired with its natural logarithm.
Products are evaluated numerically.
Column sums are displayed.
Final entropy values $e_j$ are shown.

Invariants

Same $5\times 4$ structure is used throughout.
Each criterion is processed independently by column.

Interpretation

The sequence demonstrates the mechanical steps of the entropy formula on the normalized matrix.

Table Updating with Calculated Values

Clear evidence

Shown in the video

Evidence

Animation
Observation
The table on the screen updates to show new rows for $d_j = 1 - e_j$ and $w_j$ , filling in the calculated numerical values step-by-step.

Objects

Table with criteria and calculated values

Changes

New rows for degree of diversification and weights are added
Numerical values populate the cells sequentially

Invariants

The criteria names remain unchanged

Interpretation

The visual progression demonstrates the step-by-step application of the Entropy Method formulas to the given data.

Misconceptions · 2

Confusing entropy weighting with subjective pairwise comparison

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker contrasts AHP pairwise comparison opinions with the entropy method, saying no such option is needed in entropy method.

Misconception

One might assume criterion weights always require decision-maker pairwise judgments as in AHP.

Clarification

The video states that the entropy method derives weights from the data itself and does not require pairwise comparison opinions.

Ambiguity about logarithm base

Approximate timing

Derived from the video

Evidence

Formula
Observation
Displayed formulas use $\ln$ and $h=1/\ln(m)$ .
Audio
Observation
Speaker verbally says log while the screen shows natural logarithm notation.

Uncertainties

The spoken word 'log' could be interpreted loosely, but the displayed mathematics uses ln.

Misconception

Viewers may think any logarithm base is intended because the speaker says 'log'.

Clarification

The on-screen formulas explicitly use natural logarithm $\ln$ , so the computations shown are based on $\ln$ .

Concept relations · 5

Step 1: Normalize the decision matrix → Step 2: Compute entropy for each criterion

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker says we have to calculate the normalized decision matrix first, then next we compute the entropy value.
Diagram
Observation
Step 1 precedes Step 2 in the slide sequence.

Prerequisite

Explanation

Entropy computation uses the normalized entries r_{ij}, so normalization must be completed first.

Entropy method for objective weight determination → Entropy method avoids explicit pairwise opinion elicitation

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker compares entropy method with AHP and pairwise comparison matrices.

Contrast

Explanation

The entropy method is presented as contrasting with AHP because it does not require subjective pairwise comparison inputs.

Entropy method for objective weight determination → Entropy-derived weights are called objective weights

Clear evidence

Shown in the video

Evidence

Audio
Observation
Speaker defines entropy weight and then says the resulting weights are objective weights.

Contains

Explanation

The claim that entropy-derived weights are objective weights is part of the method's introductory definition.

Entropy Formula → Degree of Diversification

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker explains that the degree of diversification is derived from the entropy value ( $1 - e_j$ ).

Prerequisite

Explanation

The entropy value must be calculated before the degree of diversification can be determined.

Degree of Diversification → Weight Vector Formula

Clear evidence

Shown in the video

Evidence

Formula
Observation
The weight formula uses ( $1 - e_j$ ), which is the degree of diversification.

Prerequisite

Explanation

The degree of diversification is a necessary component for calculating the final objective weights.

Find an answer · 7

What does entropy weight mean in multi-criteria decision making?

Clear evidence

Shown in the video

Evidence

Audio
Observation
Introductory explanation of entropy weight and objective weights.

Knowledge points

Entropy method for objective weight determination
Entropy-derived weights are called objective weights

How do you normalize a decision matrix before applying the entropy method?

Clear evidence

Shown in the video

Evidence

Formula
Observation
Normalization formula and worked table.

Knowledge points

Step 1: Normalize the decision matrix
Derivation of the normalized matrix from the example data
Mobile-phone selection example up to entropy calculation

What is the entropy formula used after normalization and what does h represent?

Clear evidence

Shown in the video

Evidence

Formula
Observation
Entropy formula and h formula displayed.

Knowledge points

Step 2: Compute entropy for each criterion
Entropy computation for the Price or Cost column
Entropy computation for the remaining criteria

What entropy values are obtained for the mobile-phone example?

Clear evidence

Shown in the video

Evidence

Diagram
Observation
Final entropy row values are shown.

Knowledge points

Mobile-phone selection example up to entropy calculation
Entropy computation for the Price or Cost column
Entropy computation for the remaining criteria

Why does the video say the entropy method does not need pairwise comparison opinions like AHP?

Clear evidence

Shown in the video

Evidence

Audio
Observation
Comparison with AHP pairwise comparison matrices.

Knowledge points

Entropy method for objective weight determination
Entropy method avoids explicit pairwise opinion elicitation

How do you calculate the final objective weights using the Entropy Method?

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker details the process of calculating the final weights using the degree of diversification.

Knowledge points

Weight Vector Formula
Derivation of Objective Weights

What is the degree of diversification in the context of the Entropy Method?

Clear evidence

Shown in the video

Evidence

Audio
Observation
The speaker explicitly defines the term 'degree of diversification'.

Knowledge points

Degree of Diversification

Coverage and review notes

Covered · Introduction to entropy-based objective weighting and contrast with AHP.

Covered · Raw $5\times 4$ mobile-phone decision matrix introduced.

Covered · Step 1 normalization formula and worked column-wise transformation.

Covered · Step 2 entropy computation with h, products, column sums, and final $e_j$ values.

Covered · No additional mathematical content beyond the already displayed entropy results.

Covered · Explanation and calculation of entropy values.

Covered · Calculation of degree of diversification and final objective weights.

Covered · Outro and call to subscribe.

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Entropy weighting ExplanationAt 0:15
Why this connection?
Reviewed explanation connection to Entropy weighting.

Questions this video answers

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Entropy Method (for weight determination of criteria/Attributes )

The explanation, unpacked.

Chapters

Learning script

Knowledge cards

Entropy method for objective weights

Step 1: Normalize the decision matrix

Step 2: Entropy formula and scaling constant

Worked entropy results for the example

Entropy Formula

Degree of Diversification

Objective Weight Calculation

Detailed learning notes

x_{ij}

r_{ij}

h

m

n

eje_jej​

∑i=1mr\sum_{i=1}^{m} r∑i=1m​r_{ij}ln⁡r\ln rlnr_{ij}

eje_jej​

h

djd_jdj​

wjw_jwj​

Entropy method for objective weight determination

Step 1: Normalize the decision matrix

Step 2: Compute entropy for each criterion

Entropy Formula

Degree of Diversification

Weight Vector Formula

Entropy-derived weights are called objective weights

Entropy method avoids explicit pairwise opinion elicitation

Derivation of the normalized matrix from the example data

Entropy computation for the Price or Cost column

Entropy computation for the remaining criteria

Derivation of Objective Weights

Mobile-phone selection example up to entropy calculation

Applying the Entropy Method to Determine Criteria Weights

Title slide introducing the topic

Raw decision matrix displayed

Visual transformation from raw values to normalized values

Visual computation of entropy column by column

Table Updating with Calculated Values

Confusing entropy weighting with subjective pairwise comparison

Ambiguity about logarithm base

Step 1: Normalize the decision matrix → Step 2: Compute entropy for each criterion

Entropy method for objective weight determination → Entropy method avoids explicit pairwise opinion elicitation

Entropy method for objective weight determination → Entropy-derived weights are called objective weights

Entropy Formula → Degree of Diversification

Degree of Diversification → Weight Vector Formula

What does entropy weight mean in multi-criteria decision making?

How do you normalize a decision matrix before applying the entropy method?

What is the entropy formula used after normalization and what does h represent?

What entropy values are obtained for the mobile-phone example?

Why does the video say the entropy method does not need pairwise comparison opinions like AHP?

How do you calculate the final objective weights using the Entropy Method?

What is the degree of diversification in the context of the Entropy Method?

Explore the knowledge in this video

Questions this video answers

What does entropy weight mean in multi-criteria decision making?+

How do you calculate the final objective weights using the Entropy Method?+

What entropy values are obtained for the mobile-phone example?+

How do you normalize a decision matrix before applying the entropy method?+

What is the entropy formula used after normalization and what does h represent?+

What is the degree of diversification in the context of the Entropy Method?+

Why does the video say the entropy method does not need pairwise comparison opinions like AHP?+

Similar explanations

$e_j$

$\sum_{i=1}^{m} r$ _{ij} $\ln r$ _{ij}

$e_j$

$d_j$

$w_j$