Shannon Entropy Method - Easy Method - How to Estimate Weights for MCDM problems #Entropy

1. Formula
   Observation

   Step 1 displays $p_{ij}=\frac{x_{ij}}{\sum_{i=1}^{m}x_{ij}}$ and the slide text says this gives the project outcomes $p_{ij}$.

2. Audio
   Observation

   The lecturer says to normalize the arrays of the decision matrix to obtain the project outcomes $p_{ij}$.

p_{ij}

Normalized project outcome or column-normalized entry obtained from the decision-matrix element $x_{ij}$.

Defined for each matrix entry by dividing $x_{ij}$ by the sum over $i=1,\dots,m$ in the same column $j$; the video does not explicitly state nonnegativity or zero-denominator handling.

1. Formula
   Observation

   The numerator and denominator in Step 1 both use $x_{ij}$.

2. Audio
   Observation

   The lecturer refers to the arrays of the decision matrix while introducing the normalization formula.

x_{ij}

Entry of the decision matrix (performance indices) at row $i$ and column $j$.

Matrix entries indexed by $i$ and $j$; the video does not separately define the full index set beyond the summation range $i=1$ to $m$.

1. Formula
   Observation

   $m$ appears as the upper limit in $\sum_{i=1}^{m}x_{ij}$ and in $k=1/\ln(m)$.

2. Audio
   Observation

   The lecturer states that $m$ is the number of alternatives which we are having in our MCDM problem.

m

Number of alternatives (rows) in the decision matrix.

Positive integer used both as the row count in normalization and inside $\ln(m)$; the video does not discuss the special case $m=1$ where $\ln(m)=0$.

1. Formula
   Observation

   Step 2 shows $E_j=-k\sum_{i=1}^{m}p_{ij}\ln p_{ij}$.

2. Audio
   Observation

   The lecturer calls this the computation of entropy measure and reads the formula with $k=1/\ln(m)$.

$E_j$

Entropy measure associated with criterion (column) $j$, computed from the normalized values $p_{ij}$ across all alternatives.

Defined for each column $j$ after normalization; the video does not explicitly state the convention for terms with $p_{ij}=0$.

1. Formula
   Observation

   The slide states 'in which $k=1/\ln(m)$'.

2. Audio
   Observation

   The lecturer says $k$ is equal to one divided by $\ln$ into $m$.

k

Scaling constant used in the entropy formula.

Set equal to $1/\ln(m)$; the video does not mention the requirement $m>1$ for this definition to be valid.

1. Formula
   Observation

   Step 3 shows $w_j=\frac{1-E_j}{\sum_{j=1}^{n}(1-E_j)}$.

2. Audio
   Observation

   The lecturer says this step defines the objective weights based on the entropy concept.

$w_j$

Objective weight assigned to criterion $j$ after entropy-based transformation.

Computed for each criterion $j$ using all $E_j$ values in the denominator sum over $j=1$ to $n$; the video does not explicitly discuss the case where all $E_j=1$.

1. Formula
   Observation

   The denominator in Step 3 is $\sum_{j=1}^{n}(1-E_j)$.

1. The symbol $n$ is visible in the formula but not verbally defined in the clip.

n

Upper index limit in the weight-normalization sum over criteria.

Appears as the number of criteria/columns by formula context; the video does not explicitly state this interpretation.

1. Formula
   Observation

   Slide shows p_{ij}=x_{ij}/\sum_{$i=1$}^{m}x_{ij}; the numerator is the current cell value used in Excel.

2. Audio
   Observation

   The speaker describes taking “the current value divided by the sum of the entire value.”

x_{ij}

Raw performance value of alternative i on criterion j in the decision matrix.

Positive numerical attribute values; in this example, material properties such as toughness index, yield strength, elastic modulus, density, thermal expansion, thermal conductivity, and specific heat.

1. Formula
   Observation

   Slide labels the normalized quantity as p_{ij}.

2. Diagram
   Observation

   Excel output block is later marked with the header “Pij”.

3. Audio
   Observation

   Speaker says the first procedure is normalizing it and later says “This is Pij.”

p_{ij}

Normalized project outcome or column-wise proportion corresponding to x_{ij}.

Values in [0,1]; each displayed column sums to 1 in the worked example.

1. Formula
   Observation

   Normalization formula uses $\sum_{i=1}^{m}x$_{ij}.

2. Audio
   Observation

   Speaker states there are seven alternatives.

3. Diagram
   Observation

   Excel table contains seven data rows for the alternatives.

m

Number of alternatives (rows) in the decision matrix.

In this example, $m=7$.

1. Formula
   Observation

   Entropy formula uses $\sum_{j=1}^{n}p$_{ij}$\ln p$_{ij}.

2. Diagram
   Observation

   Excel sheet contains seven attribute columns labeled Toughness through Specific heat.

3. Audio
   Observation

   Speaker enumerates toughness index, yield strength, elastic modulus, density, thermal expansion, thermal conductivity, and specific heat.

1. The slide notation uses j as the summation index inside $E_j$, which is notationally inconsistent with $E_j$ being indexed by a criterion; the intended meaning is summation over all alternatives for a fixed criterion.

n

Number of criteria (columns) in the decision matrix.

In this example, $n=7$.

1. Formula
   Observation

   Slide shows $E_j=-k\sum_{j=1}^{n}p$_{ij}$\ln p$_{ij}.

2. Audio
   Observation

   Speaker introduces Step 2 as computation of the entropy measure using the displayed equation.

1. The displayed formula mixes indices: $E_j$ is written with a sum over j and terms p_{ij}. The intended criterion-wise entropy should sum over alternatives i for a fixed criterion j.

$E_j$

Entropy measure associated with criterion j.

Nonnegative scalar per criterion; exact domain depends on corrected indexing.

1. Audio
   Observation

   The lecturer says Shannon entropy weight method is basically used for estimation of various weights where the decision maker is required to estimate the weights for his attributes in solving the multi criteria decision making problems.

2. Audio
   Observation

   He adds that it differs from the earlier process discussed as analytical hierarchy process because it does not consider decision variables in the same way.

The clip introduces the Shannon entropy weight method as an objective weighting procedure for multi-criteria decision making. Its role is to estimate attribute weights from data rather than relying on the earlier AHP-style subjective comparison process mentioned by the lecturer.

1. Used in multi-criteria decision making problems

2. Applied when weights for attributes/criteria must be estimated

3. Presented as different from analytical hierarchy process

1. Caption evidence
   Observation

   Slide text lists applications: social and physical sciences, economics, spectral analysis, and language modeling.

2. Audio
   Observation

   The lecturer reads these application areas and then states that C.E. Shannon proposed a mathematical theory for communication in 1948.

3. Caption evidence
   Observation

   Later bullets say entropy evaluates expected information content of a message, can be considered as a criterion for degree of uncertainty represented by a discrete probability distribution, and can be employed in decision making because it measures contrasts between sets of data and clarifies average intrinsic information transferred to the decision maker.

Entropy is presented first as a broad information-theoretic concept and then specialized to decision analysis. The video defines it as a measure of expected information content and uncertainty in a discrete probability distribution, and explains why it is useful in MCDM: it captures differences among data columns and quantifies the intrinsic information available to the decision maker.

1. Entropy is discussed in information theory

2. Uncertainty is represented by a discrete probability distribution

3. In MCDM, entropy is used to compare data sets and infer useful information for weighting

1. Caption evidence
   Observation

   Slide heading: 'Procedure for Objective weight through Shannon Entropy'.

2. Formula
   Observation

   Step 1 text: 'Normalization of the arrays of decision matrix (performance indices) to obtain the project outcomes $p_{ij}$:' followed by $p_{ij}=\frac{x_{ij}}{\sum_{i=1}^{m}x_{ij}}$.

3. Audio
   Observation

   The lecturer says the first thing is to normalize the arrays of the decision matrix to obtain the project outcomes $p_{ij}$.

The first computational step converts raw performance indices into column-normalized outcomes. Each entry $x_{ij}$ is divided by the sum of all entries in the same column, producing $p_{ij}$. This prepares the matrix for entropy calculation by turning each criterion column into a comparable set of proportions.

1. Start from a decision matrix of performance indices $x_{ij}$

2. Summation is over rows $i=1$ to $m$ within each column $j$

3. Output is the normalized matrix $p_{ij}$

1. Purpose of the Shannon entropy weight method

1. Formula
   Observation

   Step 2 text: 'Computation of the entropy measure of project outcomes using the following equation:' followed by $E_j=-k\sum_{i=1}^{m}p_{ij}\ln p_{ij}$, and 'in which $k=1/\ln(m)$.'

2. Audio
   Observation

   The lecturer says the next step is computation of entropy measure and reads the formula, specifying $k=1/\ln(m)$ where $m$ is the number of alternatives.

1. The clip does not state the convention for $p_{ij}\ln p_{ij}$ when $p_{ij}=0$.

After normalization, the method computes an entropy value $E_j$ for each criterion column $j$. The formula sums $p_{ij}\ln p_{ij}$ over all alternatives and multiplies by $-k$, with $k=1/\ln(m)$. This produces a column-level uncertainty score that will later be transformed into a weight.

1. Use the normalized values $p_{ij}$ from Step 1

2. Sum over alternatives $i=1$ to $m$ for each fixed criterion $j$

3. Set $k=1/\ln(m)$ with $m$ the number of alternatives

1. Step 1: Normalize the decision matrix
2. p_{ij}
3. m
4. k

1. Formula
   Observation

   Step 3 text: 'Defining the objective weight based on the entropy concept:' followed by $w_j=\frac{1-E_j}{\sum_{j=1}^{n}(1-E_j)}$.

2. Audio
   Observation

   The lecturer says the third step is defining the objective weights based on the entropy concept and that from the earlier step we calculate the weights.

1. The symbol $n$ is not verbally defined in the clip.

2. The clip does not discuss the degenerate case where the denominator is zero.

The final displayed step converts entropy values into criterion weights by taking the complement $1-E_j$ for each criterion and normalizing these complements across all criteria. This yields the objective weight vector used in the MCDM model.

1. Use entropy values $E_j$ from Step 2

2. Form the complement $1-E_j$ for each criterion

3. Normalize across criteria indexed by $j=1$ to $n$

1. Step 2: Compute the entropy measure for each criterion
2. $E_j$
3. n

1. Diagram
   Observation

   PowerPoint title reads “Shannon entropy Weight Method” and subtitle reads “Procedure for Objective weight through Shannon Entropy.”

2. Audio
   Observation

   Speaker explains that the method proceeds stepwise from normalization to entropy computation and then to weights.

The video presents an objective-weighting procedure for multi-criteria decision making. It starts from a raw decision matrix, normalizes each criterion column into proportions p_{ij}, computes a criterion-wise entropy $E_j$ from those proportions, and then converts entropy into weights $w_j$. In this clip, only the setup and the full normalization step are actually carried out in Excel; the entropy and weight formulas are shown but not yet computed.

1. A decision matrix with m alternatives and n criteria is available.

2. The displayed entropy formula on the slide has inconsistent indices; the corrected summation is over alternatives i for each fixed criterion j.

3. Only Step 1 is numerically executed in this clip.

1. Diagram
   Observation

   Word document shows Table 8 with materials as rows and attributes as columns.

2. Audio
   Observation

   Speaker says there are seven alternatives and lists the measured outputs.

The example uses $a 7\times 7$ decision matrix. Rows are material alternatives: Al 2024-T6, Al 5052-O, SS 301-FH, SS 310-3AH, Ti-6Al-4V, Inconel 718, and 70Cu-30Zn. Columns are criteria: Toughness index (MPa), Yield strength (MPa), Elastic modulus (GPa), Density (g/cm^3), Thermal expansion ($10^-6$/°C), Thermal conductivity (cal/s/cm^2/°C), and Specific heat (cal/g/°C). The first three are treated as MAX-type beneficial attributes and the last four as MIN-type non-beneficial attributes.

1. Rows represent alternatives.

2. Columns represent criteria.

3. Attribute direction is indicated by MAX or MIN labels in the source table.

1. Formula
   Observation

   Slide Step 1 shows p_{ij}=x_{ij}/\sum_{$i=1$}^{m}x_{ij}.

2. Audio
   Observation

   Speaker says the first procedure is normalizing it and describes dividing the current value by the sum of the entire set.

3. Animation
   Observation

   Excel formula is entered in K5 as =C5/SUM(C5:C11), then copied across columns and down rows.

Each raw entry x_{ij} is replaced by its share of the column total. For a fixed criterion j, the denominator is the sum over all m alternatives. This produces a normalized matrix P whose columns each sum to 1. In the worked example, the first computed value is 0.041518 for toughness index, obtained from 75.5 divided by the column sum of the seven toughness values.

1. Apply separately to each criterion column.

2. m is the number of alternatives.

3. The same normalization is used for all columns in this clip, without a separate transformation for MIN-type criteria.

1. Decision matrix for material selection

1. Audio
   Observation

   Speaker says the sum is common through the entire matrix and inserts a dollar symbol to fix the column throughout the part.

2. Animation
   Observation

   Formula changes from =C5/SUM(C5:C11) to =C5/SUM($C$5:$C$11).

To copy the normalization formula without changing the denominator range, the video locks the column-sum range with absolute references. This keeps the denominator fixed to the same seven cells while the numerator moves from row to row and column to column.

1. Used when copying a formula across multiple cells in Excel.

2. The locked range must correspond to the full column of raw values for the criterion being normalized.

1. Step 1: column-wise normalization to obtain p_{ij}

1. Formula
   Observation

   Slide Step 2 shows $E_j=-k\sum_{j=1}^{n}p$_{ij}$\ln p$_{ij} and $k=1/\ln(m)$.

2. Audio
   Observation

   Speaker says, “Next we will go with the second step,” and refers to computation of the entropy measure using the displayed equation.

1. The slide’s printed indices are inconsistent; the intended criterion-wise entropy should sum over $i=1$ to m for each fixed j.

2. No numerical entropy values are computed within this clip.

After normalization, the method computes an entropy value for each criterion from the normalized proportions. The slide presents the entropy expression and the constant $k=1/\ln(m)$. Conceptually, larger entropy means the criterion values are more evenly spread across alternatives, while smaller entropy indicates more discrimination among alternatives. In this clip, the formula is introduced but not yet evaluated in Excel.

1. Requires the normalized matrix p_{ij}.

2. Uses natural logarithm.

3. The displayed slide formula has a typographical/indexing error; the corrected form sums over alternatives i for each criterion j.

1. Step 1: column-wise normalization to obtain p_{ij}
2. Shannon entropy weight method overview

1. Formula
   Observation

   Slide Step 3 shows $w_j=(1-E_j)/\sum_{j=1}^{n}(1-E_j)$.

2. Audio
   Observation

   Speaker mentions defining the objective weight based on the entropy concept.

1. The weight formula is visible but not applied numerically in this clip.

The final step converts each criterion’s entropy into a weight by taking $1-E_j$ and normalizing these quantities across all criteria. This yields weights that sum to 1. Criteria with lower entropy receive larger weights because they provide more distinguishing information across alternatives.

1. Requires entropy values $E_j$ for all criteria.

2. Weights are normalized to sum to 1.

3. Not numerically executed within this clip.

1. Step 2: entropy measure of project outcomes

1. Diagram
   Observation

   PowerPoint title reads "Shannon entropy Weight Method" and subtitle reads "Procedure for Objective weight through Shannon Entropy".

2. Formula
   Observation

   Slide lists Step 1 normalization and Step 2 entropy computation.

The clip presents a two-step procedure for objective weighting: first normalize the decision-matrix entries into p_{ij}, then compute an entropy measure $E_j$ for each criterion using the normalized values and the constant $k=1/\ln(m)$.

1. A decision matrix with entries x_{ij} is available.

2. There are m rows (alternatives) and multiple criteria columns.

3. The entropy step is applied column-wise to the normalized values.

1. Caption evidence
   Observation

   Slide bullet: 'A mathematical theory of communication was proposed by C.E. Shannon (1948).'

2. Audio
   Observation

   The lecturer says this was proposed by C.E. Shannon, a mathematical theory for communication in the year 1948.

A mathematical theory of communication was proposed by C.E. Shannon in 1948.

Historical factual statement about attribution and year.

1. Caption evidence
   Observation

   Slide bullet: 'Entropy concept in information theory can be considered as a criterion for the degree of uncertainty represented by a discrete probability distribution.'

2. Audio
   Observation

   The lecturer reads essentially the same statement aloud.

In information theory, entropy can be considered as a criterion for the degree of uncertainty represented by a discrete probability distribution.

1. The setting is information theory.

2. Uncertainty is represented by a discrete probability distribution.

General conceptual claim stated without formal proof in the clip.

1. Caption evidence
   Observation

   Slide bullet: 'Entropy idea can be effectively employed in the process of decision making, because it measures existent contrasts between sets of data and clarifies the average intrinsic information transferred to decision maker.'

2. Audio
   Observation

   The lecturer reads this rationale aloud.

Entropy can be effectively employed in decision making because it measures existing contrasts between sets of data and clarifies the average intrinsic information transferred to the decision maker.

1. There are multiple sets of data to compare.

2. The decision context requires extracting informative differences among criteria.

Conceptual justification stated in the lecture, not proved within the clip.

1. Diagram
   Observation

   Excel displays a completed $7\times 7$ normalized block under the label Pij.

2. Formula
   Observation

   Normalization formula divides each entry by its column total.

For every criterion j, $\sum_{i=1}^{m}p$_{ij}=1 after applying p_{ij}=x_{ij}/\sum_{$i=1$}^{m}x_{ij}.

1. The denominator is the sum of all m raw values in the same column.

2. All x_{ij} are finite and the column sum is nonzero.

For each column j and all rows $i=1$,$\dots$,m.

1. Formula
   Observation

   Slide prints $E_j=-k\sum_{j=1}^{n}p$_{ij}$\ln p$_{ij}.

2. Formula
   Observation

   The left-hand side is indexed by j while the summation also uses j, and the summand still contains i.

As printed, the slide’s entropy formula is notationally inconsistent; the standard criterion-wise form should sum over alternatives i for a fixed criterion j.

1. The intended method is Shannon entropy weighting by criterion.

2. $E_j$ denotes entropy of criterion j.

For each criterion j, the sum should run over $i=1$,$\dots$,m.

1. Formula
   Observation

   Three successive slides display Step 1, Step 2, and Step 3 formulas in order.

2. Audio
   Observation

   The lecturer narrates the sequence: first normalize, next compute entropy measure, and the third step define objective weights.

1. Expression

   $$p_{ij}=\frac{x_{ij}}{\sum_{i=1}^{m}x_{ij}}$$
   Explanation

   Normalize each decision-matrix entry by the sum of its column to obtain $p_{ij}$.

   Justification

   This is the Step 1 formula shown on the slide and read by the lecturer.

   Shown in the video
2. Expression

   $$E_j=-k\sum_{i=1}^{m}p_{ij}\ln p_{ij},\quad k=\frac{1}{\ln(m)}$$
   Explanation

   For each criterion $j$, compute an entropy value from the normalized column values.

   Justification

   This is the Step 2 formula shown on the slide; the lecturer identifies $m$ as the number of alternatives.

   Shown in the video
3. Expression

   $$w_j=\frac{1-E_j}{\sum_{j=1}^{n}(1-E_j)}$$
   Explanation

   Convert entropy values into weights by taking complements and normalizing across all criteria.

   Justification

   This is the Step 3 formula shown on the slide and described as defining the objective weights.

   Shown in the video

The Shannon entropy weight method proceeds from raw decision-matrix entries $x_{ij}$ to normalized outcomes $p_{ij}$, then to criterion entropies $E_j$, and finally to objective weights $w_j$.

1. Audio
   Observation

   Speaker says to take the current value divided by the sum of the entire value and then fix the column with a dollar symbol.

2. Animation
   Observation

   Excel shows =C5/SUM(C5:C11), then =C5/SUM($C$5:$C$11), producing 0.041518.

1. Expression

   $$p_{1,\text{Toughness}}=\frac{x_{1,\text{Toughness}}}{\sum_{i=1}^{7}x_{i,\text{Toughness}}}$$
   Explanation

   Write the normalization rule for the first toughness entry.

   Justification

   Definition of Step 1 from the slide.

   Shown in the video
2. Expression

   $$=C5/SUM(C5:C11)$$
   Explanation

   Enter the raw value divided by the column sum in Excel.

   Justification

   Direct implementation of the normalization formula.

   Shown in the video
3. Expression=C5/SUM($C$5:$C$11)
   Explanation

   Lock the denominator range so it stays fixed when the formula is copied.

   Justification

   Excel absolute referencing preserves the same column total across rows and columns.

   Shown in the video
4. Expression

   $$0.041518$$
   Explanation

   The computed normalized value appears in the cell.

   Justification

   Numerical evaluation of the formula in the spreadsheet.

   Shown in the video

The first normalized entry is obtained by dividing the raw toughness value by the total toughness column sum, and absolute references allow the same rule to be copied consistently.

1. Animation
   Observation

   The formula is dragged across columns K to Q and then down through seven rows.

2. Diagram
   Observation

   A full $7\times 7$ block of decimals appears and is later labeled Pij.

1. Expression

   $$K5:Q5$$
   Explanation

   Copy the first-row normalization formula across all seven criteria.

   Justification

   Each column has its own fixed denominator range once absolute references are used.

   Shown in the video
2. Expression

   $$K5:Q11$$
   Explanation

   Drag the filled first row downward to cover all seven alternatives.

   Justification

   The same column-wise normalization applies to every row.

   Shown in the video
3. Expression

   $$P_{ij}$$
   Explanation

   Label the resulting block as the normalized matrix.

   Justification

   These entries correspond exactly to p_{ij} from Step 1.

   Shown in the video

The complete normalized matrix P is produced by extending the single-cell formula across all criteria columns and all alternative rows.

1. Audio
   Observation

   Speaker says first we need to know about m, m is equal to seven, then k is equal to one divided by log of m.

2. Diagram
   Observation

   Excel shows m entered as 7 and k computed as 1.183294662.

1. The exact Excel base argument inside LOG is not fully legible in every frame.

1. Expression

   $$m=7$$
   Explanation

   The speaker identifies the number of alternatives from the data table.

   Justification

   Seven rows are present in the Excel matrix and the speaker explicitly says m is equal to seven.

   Shown in the video
2. Expression

   $$k=\frac{1}{\ln(m)}$$
   Explanation

   The slide formula defines the scaling constant.

   Justification

   Directly read from the PowerPoint equation.

   Shown in the video
3. Expression

   $$k=\frac{1}{\ln(7)}\approx 1.183294662$$
   Explanation

   Substituting $m=7$ gives the numerical constant used later.

   Justification

   Excel displays the computed value 1.183294662 after the formula entry.

   Shown in the video

For this example, k is approximately 1.183294662.

1. Audio
   Observation

   Speaker says they will calculate p_{ij} into log of p_{ij} and drag it vertically down up to seven.

2. Diagram
   Observation

   Excel formula bar shows =K4*LOG(K4) and the resulting negative values are filled down the column.

1. The spoken variable name is unclear and sounds like "p a j" or "n a j" in places.

1. Expression

   $$p_{ij}\ln p_{ij}$$
   Explanation

   Each normalized entry is multiplied by its logarithm.

   Justification

   This is the inner term of the entropy formula shown on the slide.

   Shown in the video
2. Expression

   $$=K4*LOG(K4)$$
   Explanation

   In Excel, the first product is entered directly for the top cell of the normalized column.

   Justification

   Visible in the formula bar during the demonstration.

   Shown in the video
3. Expression

   $$\text{fill down for }i=1,\dots,7$$
   Explanation

   The same product formula is copied down through all seven alternatives.

   Justification

   The speaker says he will drag it vertically down up to seven and the grid fills accordingly.

   Shown in the video

A column of values p_{ij}$\ln p$_{ij} is obtained for each criterion.

1. Audio
   Observation

   Speaker says summation has to be done after multiplying p_{ij} with the log term and then estimates the sum value.

2. Diagram
   Observation

   Excel formula bar shows =SUM(K12:K18) and the result -0.72253.

1. Expression

   $$\sum_{i=1}^{m}p_{ij}\ln p_{ij}$$
   Explanation

   The entropy formula requires summing the product terms across all alternatives in a column.

   Justification

   Read from the slide equation.

   Shown in the video
2. Expression

   $$=SUM(K12:K18)$$
   Explanation

   The speaker applies Excel's SUM function to the seven product cells in one column.

   Justification

   Visible in the formula bar and echoed by the narration.

   Shown in the video
3. Expression

   $$-0.72253$$
   Explanation

   For the displayed column, the sum of products is negative.

   Justification

   Excel shows -0.72253 as the computed sum.

   Shown in the video

The column-wise sum of p_{ij}$\ln p$_{ij} is computed before applying the -k factor.

1. Audio
   Observation

   Speaker says the formula here is sum multiplied by minus k, adds dollar signs so k stays fixed, drags the formula, reduces to three digits, and labels the row Eij.

2. Diagram
   Observation

   Excel formula bar shows =-($I13*K20$) and later =-($I$$13*K20$); the resulting row is labeled Eij with values 0.855, 0.879, 0.954, 0.953, 0.979, 0.819, 0.964.

1. The exact cell references vary by column and are only partially legible; the structural formula is clear.

1. Expression

   $$E_j=-k\sum_{i=1}^{m}p_{ij}\ln p_{ij}$$
   Explanation

   The slide formula states that the entropy measure equals minus k times the column sum.

   Justification

   Directly from the PowerPoint equation.

   Shown in the video
2. Expression

   $$=-(I13*K20)$$
   Explanation

   In Excel, the speaker multiplies the negative of the stored sum by the stored k value.

   Justification

   Visible in the formula bar during entry.

   Shown in the video
3. Expression=-($I$$13*K20$)
   Explanation

   Dollar signs are added so the k reference remains fixed when the formula is dragged across columns.

   Justification

   Speaker explicitly says k will be constant and adds the dollar symbol so row and column will be fixed.

   Shown in the video
4. Expression

   $$\text{round/display to three decimals}$$
   Explanation

   The resulting entropy values are shown with three decimal places.

   Justification

   Speaker says he will reduce it to three digits and the displayed row shows three-decimal values.

   Shown in the video

The final row labeled Eij contains the entropy measures for the criteria, here displayed as 0.855, 0.879, 0.954, 0.953, 0.979, 0.819, 0.964.

1. Formula
   Observation

   Slide Step 3 shows $w_j=(1-E_j)/\sum_{j=1}^{n}(1-E_j)$.

2. Audio
   Observation

   The speaker explains the sequence: estimate EIJ, compute 1 minus EIJ, take the sum, then divide each value by the summation.

3. Diagram
   Observation

   Excel implements the sequence with rows Eij, 1-Eij, the total 0.597, and the final wij row.

1. Expression

   $$w_j=\frac{1-E_j}{\sum_{j=1}^{n}(1-E_j)}$$
   Explanation

   Start from the Step 3 formula shown on the slide for defining the objective weight based on entropy.

   Justification

   Directly observed from the slide formula and repeated in the narration.

   Shown in the video
2. Expression

   $$1-E_j$$
   Explanation

   For each criterion, compute the complement of its entropy value.

   Justification

   The speaker says this is the first thing needed, and Excel creates the row 1-Eij with formulas such as =1-K22.

   Shown in the video
3. Expression

   $$\sum_{j=1}^{n}(1-E_j)$$
   Explanation

   Sum all the complements across the criteria to form the common denominator.

   Justification

   The speaker explicitly says the sum is needed before estimating wij, and Excel evaluates =SUM(K24:Q24)=0.597.

   Shown in the video
4. Expression

   $$w_j=\frac{1-E_j}{0.597}$$
   Explanation

   Divide each individual complement by the total sum to obtain the normalized weight for that criterion.

   Justification

   The spreadsheet formula =K24/$SR$24 is entered and dragged across the row, matching the slide formula.

   Shown in the video
5. Expression

   $$\sum_{j=1}^{n} w_j = 1$$
   Explanation

   The resulting weights form a normalized vector whose entries add to 1.

   Justification

   The speaker states that adding all the weights gives 1, using this as the check on the procedure.

   Shown in the video

The clip derives the objective weight vector by converting entropy values into complements, summing those complements, and normalizing each complement by that total so that the final weights sum to 1.

1. Caption evidence
   Observation

   A PDF page appears with the caption 'Table 8 Decision matrix for Example 2 (Manshadi et al. 2007)'.

2. Audio
   Observation

   The lecturer says he will go through one solved paper taken from the NCA journal and solve it using Excel software to estimate the weights.

3. Audio
   Observation

   He further says this is a decision matrix solved by Manshadi for material selection for a cryogenic tank.

1. The spoken source name sounds like 'NCA journal' but is not visually confirmed in the clip.

2. No numerical computation is actually carried out before the clip ends.

Use a published decision matrix to illustrate the Shannon entropy weighting procedure on a real MCDM dataset.

1. A PDF table labeled 'Table 8 Decision matrix for Example 2 (Manshadi et al. 2007)' is shown.

2. The table lists materials and seven attributes: Toughness index (MPa), Yield strength (MPa), Elastic modulus (GPa), Density (g/cm^3), Thermal expansion ($10^-6$/°C), Thermal conductivity (cal/s/cm/°C), Specific heat (cal/g/°C).

3. Attribute directions are marked as MAX for the first three and MIN for the last four.

4. The lecturer states the example concerns material selection for a cryogenic tank.

Set up the example data that will later be processed by the three-step entropy method.

1. Expression

   $$$$
   Explanation

   The lecturer announces that he will use a solved paper and Excel software to estimate the weights.

   Justification

   This is stated directly in the audio near the transition from slides to the PDF.

   Shown in the video
2. Expression

   $$$$
   Explanation

   The screen changes to a PDF page showing Table 8, which provides the raw decision matrix for the example.

   Justification

   The visual evidence is the table caption and the visible matrix structure.

   Shown in the video
3. Expression

   $$$$
   Explanation

   The lecturer identifies the table as a decision matrix from Manshadi for material selection for a cryogenic tank.

   Justification

   This identification comes from the spoken narration accompanying the table display.

   Shown in the video

The clip only introduces the example and displays the source decision matrix; no weights are computed within the provided 180 seconds.

Verification would require applying the displayed Step 1–Step 3 formulas to the table data, but those calculations are not shown in this fragment.

1. Diagram
   Observation

   Word document shows Table 8 with seven materials and seven attributes.

2. Audio
   Observation

   Speaker says there are seven alternatives and lists the measured outputs.

Set up a multi-criteria material selection problem with seven alternatives and seven attributes before applying the Shannon entropy weight method.

1. Alternatives: Al 2024-T6, Al 5052-O, SS 301-FH, SS 310-3AH, Ti-6Al-4V, Inconel 718, 70Cu-30Zn.

2. Criteria: Toughness index (MPa), Yield strength (MPa), Elastic modulus (GPa), Density (g/cm^3), Thermal expansion ($10^-6$/°C), Thermal conductivity (cal/s/cm^2/°C), Specific heat (cal/g/°C).

3. First three criteria are labeled MAX; last four are labeled MIN.

Identify the structure of the decision matrix that will be normalized in Step 1.

1. Expression

   $$7\times 7\text{ matrix}$$
   Explanation

   Arrange materials as rows and attributes as columns.

   Justification

   This is the standard decision-matrix layout shown in the source table.

   Shown in the video
2. Expression

   $$\text{MAX/MIN labels}$$
   Explanation

   Record whether each criterion is beneficial or non-beneficial.

   Justification

   The table explicitly marks attribute direction.

   Shown in the video

A 7-by-7 decision matrix of material alternatives versus engineering criteria is prepared for entropy weighting.

The matrix dimensions match the spoken count of seven alternatives and the seven visible attribute columns.

1. Animation
   Observation

   Excel formula bar shows =C5/SUM($C$5:$C$11) and result 0.041518.

2. Audio
   Observation

   Speaker explains current value divided by the sum of the entire set.

Compute the normalized value $p_{1,\text{Toughness}}$ for the first material in the toughness column.

1. Raw first toughness value is in cell C5.

2. The toughness column occupies C5:C11.

3. Normalization formula is p_{ij}=x_{ij}/\sum_{$i=1$}^{m}x_{ij}.

Obtain the first entry of the normalized matrix.

1. Expression

   $$=C5/SUM(C5:C11)$$
   Explanation

   Divide the first raw value by the column total.

   Justification

   Direct substitution into the normalization formula.

   Shown in the video
2. Expression=C5/SUM($C$5:$C$11)
   Explanation

   Make the denominator absolute so the same column total is used when copying.

   Justification

   Excel absolute referencing fixes the range.

   Shown in the video
3. Expression

   $$0.041518$$
   Explanation

   Read the resulting normalized value from the cell.

   Justification

   Spreadsheet computation.

   Shown in the video

0.041518

The value is displayed directly in the Excel cell after formula entry.

1. Animation
   Observation

   The formula is copied across K:Q and down seven rows.

2. Diagram
   Observation

   The completed decimal block is labeled Pij.

Extend the single normalization formula to all seven criteria and seven alternatives.

1. Raw data occupy columns C:I and rows 5:11.

2. The first normalized formula has already been entered in K5.

3. Each criterion column must be normalized independently.

Produce the complete matrix of p_{ij} values.

1. Expression

   $$K5:Q5$$
   Explanation

   Copy the formula horizontally across all criterion columns.

   Justification

   Each column has its own fixed denominator range.

   Shown in the video
2. Expression

   $$K5:Q11$$
   Explanation

   Copy the first normalized row vertically through all alternatives.

   Justification

   The same column-wise rule applies to every row.

   Shown in the video
3. Expression

   $$Pij$$
   Explanation

   Label the resulting block as the normalized matrix.

   Justification

   These entries are exactly the p_{ij} values from Step 1.

   Shown in the video

$A 7\times 7$ matrix of normalized proportions is obtained and labeled Pij.

The visible Excel block contains seven rows and seven columns of decimal values, matching the stated dimensions.

1. Diagram
   Observation

   Excel worksheet shows a decision matrix with criteria headings and a normalized p_{ij} block, followed by computed product, sum, and entropy rows.

2. Audio
   Observation

   Speaker narrates entering $m=7$, computing k, forming p_{ij}$\ln p$_{ij}, summing, multiplying by -k, and labeling the result Eij.

1. Some intermediate cell references and a few decimal entries are too small to verify exactly from the frames.

Given a decision matrix with $m=7$ alternatives and several criteria, compute the entropy measure row using the Shannon entropy weight method.

1. $m=7$.

2. Normalized matrix entries p_{ij} are already prepared in Excel.

3. $k=1/\ln(m)$.

4. Entropy formula is $E_j=-k\sum_{i=1}^{m}p$_{ij}$\ln p$_{ij}.

Compute k, form the products p_{ij}$\ln p$_{ij}, sum them by column, and obtain the entropy row $E_j$.

1. Expression

   $$m=7$$
   Explanation

   The number of alternatives is read from the table size.

   Justification

   Seven data rows are visible and the speaker states m is equal to seven.

   Shown in the video
2. Expression

   $$k=\frac{1}{\ln(7)}\approx 1.183294662$$
   Explanation

   The scaling constant is calculated once for the whole worksheet.

   Justification

   Excel shows the computed k value after the formula entry.

   Shown in the video
3. Expression

   $$=K4*LOG(K4)$$
   Explanation

   For each normalized entry, multiply the entry by its logarithm.

   Justification

   This implements the inner term p_{ij}$\ln p$_{ij} from the slide formula.

   Shown in the video
4. Expression

   $$=SUM(K12:K18)$$
   Explanation

   Sum the seven product values in a criterion column.

   Justification

   This gives $\sum_{i=1}^{m}p$_{ij}$\ln p$_{ij} for that column.

   Shown in the video
5. Expression=-($I$$13*K20$)
   Explanation

   Multiply the column sum by -k, keeping k fixed with absolute referencing.

   Justification

   This matches $E_j=-k\sum_{i=1}^{m}p$_{ij}$\ln p$_{ij} and allows dragging across columns.

   Shown in the video
6. Expression

   $$Eij: 0.855, 0.879, 0.954, 0.953, 0.979, 0.819, 0.964$$
   Explanation

   The resulting entropy values are displayed to three decimal places and labeled Eij.

   Justification

   Visible in the final Excel row and described by the speaker.

   Shown in the video