Multi Criteria Decision Making - Example

1. Formula
   Observation

   The slide shows the notation X_{ij} = Performance Value of i^{th} Alternative over j^{th} criteria.

2. Audio
   Observation

   The narrator explains that each value in each cell is called the performance value and gives the example that 300 is the performance value of the third alternative over the first criterion.

X_{ij}

Performance value of the i-th alternative with respect to the j-th criterion.

i indexes alternatives (Mobile 1 to Mobile 5); j indexes criteria (Price or Cost, Storage Space, Camera, Looks).

1. Formula
   Observation

   Appears in X_{ij} = Performance Value of i^{th} Alternative over j^{th} criteria.

2. Audio
   Observation

   The narrator refers to the third alternative as Mobile 3 when explaining a cell value.

i

Index of an alternative (row) in the decision matrix.

Rows labeled Mobile 1 through Mobile 5.

1. Formula
   Observation

   Appears in X_{ij} = Performance Value of i^{th} Alternative over j^{th} criteria.

2. Audio
   Observation

   The narrator refers to the first criterion as cost when explaining the example value 300.

j

Index of a criterion (column) in the decision matrix.

Columns labeled Price or Cost, Storage Space, Camera, and Looks.

1. Formula
   Observation

   Bottom formulas use $X_{ij}$ in both normalization rules: Non Beneficial = Min(X_ij)/X_ij and Beneficial = X_ij/Max(X_ij).

1. The video does not explicitly define the row index i or column index j.

X_{ij}

Performance value in an individual cell of the decision matrix for a given alternative and criterion.

Positive numeric attribute values shown in the table; units differ by criterion (dollars, GB, MP, or unitless rating).

1. Formula
   Observation

   Formula text shows Min(X_ij) in the non-beneficial normalization rule.

2. Audio
   Observation

   Narrator says that for non-beneficial criteria we take the minimum value.

$\operatorname{Min}$(X_{ij})

Minimum value among the entries of one criterion column.

Column-wise over all alternatives for the same criterion.

1. Formula
   Observation

   Formula text shows Max(X_ij) in the beneficial normalization rule.

2. Audio
   Observation

   Narrator says that for beneficial criteria we divide by the maximum value.

$\operatorname{Max}$(X_{ij})

Maximum value among the entries of one criterion column.

Column-wise over all alternatives for the same criterion.

1. Diagram
   Observation

   Top row labeled Weightage appears above the four criteria columns.

2. Audio
   Observation

   Narrator says the next step is to assign the weightage to the criteria.

1. The video does not show a formula using the weights yet.

$\text{Weightage}$

Relative importance assigned to each criterion before computing a final score.

Shown as percentages 25% for each of the four criteria.

1. Diagram
   Observation

   The top row of the table is labeled Weightage and shows 25% under each of the four criteria.

2. Audio
   Observation

   The speaker says the sum of weightage is always 100 or 100 percent.

25%

Percentage weight assigned to each criterion in the example.

Criterion weights expressed as percentages; in this example each of the four criteria has 25%, and the total is 100%.

1. Diagram
   Observation

   The same Weightage row changes from 25% to 0.25 for all four criteria.

2. Audio
   Observation

   The speaker says to convert the percent into number and then multiply.

0.25

Decimal form of each criterion weight used in multiplication.

Dimensionless decimal weights in [0,1]; here each criterion weight equals 0.25.

1. Diagram
   Observation

   The body of the matrix contains values such as 0.8, 0.5, 0.75, 1, 0.667, 0.727, and 0.889 for Mobile 1 through Mobile 5 across Price or Cost, Storage Space, Camera, and Looks.

2. Audio
   Observation

   The speaker refers to multiplying the weight assigned to each criteria with its normalized performance value.

1. The video does not show how these normalized performance values were obtained from raw data.

normalized performance value

The entry in the decision matrix for a given alternative and criterion before weighting.

Displayed as decimal scores in the matrix cells; the example uses values between 0 and 1.

1. Diagram
   Observation

   Cells are rewritten as products such as 0.8 x 0.25, 0.5 x 0.25, 0.75 x 0.25, and 1 x 0.25, then replaced by results such as 0.2, 0.125, 0.1875, and 0.25.

2. Audio
   Observation

   The speaker says on solving we get the weighted normalized decision matrix.

weighted normalized entry

Product of a criterion weight and the corresponding normalized performance value.

Computed cell values in the weighted normalized decision matrix.

1. Diagram
   Observation

   A new rightmost column labeled Performance Score appears with values 0.7625, 0.65, 0.8667, 0.7067, and 0.6972. The displayed Mobile 4 total 0.7067 is an arithmetic error; its displayed components total 0.7567.

2. Audio
   Observation

   The speaker says to add all weighted normalized performance value of each alternative to get a performance score.

Performance Score

Row sum of the weighted normalized entries for one alternative.

Scalar summary score per alternative; in this example the displayed scores range from 0.65 to 0.8667.

1. Audio
   Observation

   The narrator says that in the previous video they dealt with the introduction of MCDM, that is Multi Criteria Decision Making, and were dealing with the example of selecting the best mobile phone among all available alternatives based on various criteria like cost, storage space, camera quality, looks.

2. Diagram
   Observation

   The slide title reads Selection of the Best, and the table lists five mobile alternatives across four criteria.

This clip frames multi-criteria decision making as choosing the best alternative from several options using multiple criteria. The concrete example is selecting among five mobile phones using the criteria Price or Cost, Storage Space, Camera, and Looks.

1. There are multiple alternatives.

2. There are multiple criteria for evaluating those alternatives.

3. The goal is selection of the best alternative.

1. Audio
   Observation

   The narrator states that all criteria have different units: cost is represented in dollar, storage space in gigabyte, camera quality in megapixel, so we cannot compare cost with storage space or camera quality.

2. Diagram
   Observation

   The bottom text box reads SOME ISSUES IN SELECTING THE BEST with the bullet All Criteria have different Unit.

3. Diagram
   Observation

   The table entries visibly use mixed units such as 250 $, 16 GB, and 12 MP.

Before comparing alternatives across criteria, the clip identifies a problem: the criteria are not commensurable because they use different measurement units. Cost uses dollars, storage uses gigabytes, and camera quality uses megapixels, so direct comparison across these columns is not meaningful without normalization or conversion.

1. The decision matrix contains criteria with heterogeneous units.

2. Direct cross-criterion comparison is not yet valid.

1. MCDM example setup: selecting the best mobile phone

1. Audio
   Observation

   The narrator says another issue is the use of linguistic term; we do not express looks or appearance of mobile in some unit, we say it's good looking or it looks excellent.

2. Diagram
   Observation

   The bottom text box includes the bullet Use of Linguistic terms.

3. Diagram
   Observation

   The Looks column contains words such as Excellent, Average, Good, and Below Average.

A second obstacle is that one criterion, Looks, is expressed with verbal labels rather than numerical measurements. Terms such as Excellent, Good, Average, and Below Average are qualitative and must be converted into numbers before they can be handled like the other matrix entries.

1. At least one criterion is qualitative rather than numeric.

2. The raw table contains linguistic labels instead of measurable values.

1. MCDM example setup: selecting the best mobile phone

1. Audio
   Observation

   The narrator says this issue can be resolved by using a conversion scale, and here a five point scale is used in which low corresponds to one and excellent corresponds to five.

2. Diagram
   Observation

   An overlay titled Using 5 point Scale appears with rows Low=1, Below Average=2, Average=3, Good=4, Excellent=5.

1. The spoken explanation explicitly names only the endpoints low=1 and excellent=5; the intermediate mappings are visible on the slide but not fully spoken in the audio.

To handle qualitative ratings, the clip introduces a 5-point scale that maps linguistic labels to integers. The displayed mapping is Low → 1, Below Average → 2, Average → 3, Good → 4, Excellent → 5. This converts the Looks column into numerical form compatible with the rest of the matrix.

1. The linguistic labels are ordered from worse to better.

2. A common numeric scale is chosen consistently for all qualitative entries.

1. Issue 2: qualitative linguistic ratings

1. Audio
   Observation

   The narrator says after substituting the numerical value we get a decision matrix, and in this each value in each cell is called or known as the performance value.

2. Diagram
   Observation

   The bottom label changes to Decision Matrix, and the formula line reads X_{ij} = Performance Value of i^{th} Alternative over j^{th} criteria.

3. Diagram
   Observation

   The Looks column now shows numeric values 5, 3, 4, 4, 2 instead of words.

After replacing linguistic terms with numbers, the table is identified as a decision matrix. Each entry X_{ij} is the performance value of alternative i under criterion j. In the example, 300 is described as the performance value of Mobile 3 for the first criterion, cost.

1. All criterion entries are expressed numerically.

2. Rows correspond to alternatives and columns correspond to criteria.

1. Five-point conversion scale for linguistic terms
2. X_{ij}

1. Audio
   Observation

   The narrator says next we have to categorize the criteria into non-beneficial and beneficial criteria, and non-beneficial criteria are those criteria whose lower value is desired, for example cost.

2. Diagram
   Observation

   Orange headers appear above the columns: Non Beneficial over Price or Cost, and Beneficial over Storage Space, Camera, and Looks.

1. The clip defines non-beneficial criteria explicitly, but it does not give a full spoken definition of beneficial criteria before ending.

The next step is to classify each criterion by whether larger or smaller values are preferred. The slide marks Price or Cost as Non Beneficial and the other three columns as Beneficial. The narration explicitly defines non-beneficial criteria as those whose lower value is desired, using cost as the example.

1. Each criterion must be assigned a preference direction.

2. Non-beneficial means lower numerical values are better.

1. Decision matrix and performance value

1. Diagram
   Observation

   Slide title reads Selection of the Best.

2. Diagram
   Observation

   Table lists Mobile 1 through Mobile 5 with columns Price or Cost, Storage Space, Camera, Looks.

3. Audio
   Observation

   Narrator explains that lowest cost is desired and higher values are desired for storage space, camera, and looks.

This segment introduces a multi-criteria selection problem using a decision table. The rows are alternatives (Mobile 1 to Mobile 5), and the columns are criteria. The example distinguishes between criteria where lower values are better and criteria where higher values are better.

1. The table is used to compare several alternatives across multiple attributes.

2. Each criterion is classified as either beneficial or non-beneficial.

1. Diagram
   Observation

   Header labels mark Price or Cost as Non Beneficial and Storage Space, Camera, Looks as Beneficial.

2. Audio
   Observation

   Narrator says beneficial criteria are those whose higher value is desired.

A beneficial criterion is one for which a larger numerical value is preferred. A non-beneficial criterion is one for which a smaller numerical value is preferred. In this example, Price or Cost is non-beneficial, while Storage Space, Camera, and Looks are beneficial.

1. Applies when comparing alternatives on mixed-direction attributes.

2. The classification depends on whether larger or smaller raw values are desirable.

1. Selection of the Best decision table

1. Audio
   Observation

   Narrator says, In order to make all criteria comparable, we do normalization.

2. Diagram
   Observation

   The word Normalization appears at upper left once the formulas are introduced.

Normalization is presented as the step that makes different criteria comparable before weighting and ranking. It converts raw values with different units and directions into a common scale.

1. Used after identifying beneficial and non-beneficial criteria.

2. Needed before combining criteria into a single comparison.

1. Beneficial versus non-beneficial criteria

1. Formula
   Observation

   Bottom-right formula reads Beneficial- X_ij / Max(X_ij).

2. Audio
   Observation

   Narrator says that in beneficial criteria we divide the performance value in individual cell with the maximum value.

3. Animation
   Observation

   Storage Space, Camera, and Looks columns are rewritten as fractions with the column maximum in the denominator.

For a beneficial criterion, each cell value is divided by the maximum value in that criterion column. This preserves the preference direction because the best raw value becomes 1 and smaller values become fractions below 1.

1. Applies only to beneficial criteria.

2. The maximum is taken within the same criterion column.

3. All entries in the column must be positive for the displayed division method.

1. Purpose of normalization
2. X_{ij}
3. $\operatorname{Max}$(X_{ij})

1. Formula
   Observation

   Bottom-left formula reads Non Beneficial- Min(X_ij) / X_ij.

2. Audio
   Observation

   Narrator says that for non-beneficial criteria we take the minimum value and divide it with the performance value of each cell.

3. Animation
   Observation

   Price or Cost column is rewritten as $200/250$, $200/200$, $200/300$, $200/275$, $200/225$.

For a non-beneficial criterion, the minimum value in the column is placed in the numerator and each cell value is placed in the denominator. This reverses the direction so that the smallest raw value becomes 1 and larger costs become fractions below 1.

1. Applies only to non-beneficial criteria.

2. The minimum is taken within the same criterion column.

3. Entries must be nonzero and positive for the displayed division method.

1. Purpose of normalization
2. X_{ij}
3. $\operatorname{Min}$(X_{ij})

1. Diagram
   Observation

   Fractions disappear and the table fills with decimal values.

2. Diagram
   Observation

   Label Normalized Decision Matrix appears below the table.

3. Audio
   Observation

   Narrator says, On solving, we will get a normalized decision matrix.

After applying the two normalization rules to every column, the original decision table becomes a normalized decision matrix. Each entry is now a dimensionless number on a comparable scale, ready for weighting.

1. Obtained by normalizing every criterion column according to its type.

2. Values shown in the example lie between 0 and 1.

1. Beneficial normalization formula
2. Non-beneficial normalization formula

1. Audio
   Observation

   The narrator states: Non-beneficial criteria are those criteria whose lower value is desired, for example cost.

2. Diagram
   Observation

   The header Non Beneficial is placed above the Price or Cost column.

A criterion is non-beneficial if a lower value is desired.

1. The criterion has been numerically represented in the decision matrix.

For the criteria under consideration in this example.

1. Audio
   Observation

   The narrator says cost is represented in dollar, storage space in gigabyte, camera quality in megapixel, so we cannot compare cost with storage space or camera quality.

2. Diagram
   Observation

   The bullet All Criteria have different Unit is shown beneath the table.

If criteria are measured in different units, their raw values cannot be directly compared with one another.

1. The criteria retain their original heterogeneous units.

For the listed criteria in the example.

1. Audio
   Observation

   For some selfie lover, camera will have a higher weightage. They may assign the camera a weightage of around 50 to 60 percent.

Different users may assign different importance to criteria; for a selfie lover, the camera criterion may receive about 50% to 60% weightage.

1. The decision problem involves multiple criteria whose relative importance can differ by person.

Existential example: some users may assign higher weight to camera.

1. Audio
   Observation

   the sum of weightage is always 100 or 100 percent

2. Diagram
   Observation

   Four 25% weights are shown together with 100% at the right.

The sum of all criterion weightages is 100%.

1. The listed criteria exhaust the weights used in the decision matrix.

Universal over the criteria in the example.

1. Audio
   Observation

   based on the weightage assigned to each criteria, mobile three is the best alternative among all other alternative as it is having rank one.

2. Diagram
   Observation

   Mobile 3 has performance score 0.8667 and Rank 1.

Under the stated equal weights, Mobile 3 is the best alternative because it has the highest performance score and rank 1.

1. The normalized performance values in the table are accepted as given.

2. Each criterion has weight 0.25.

Comparative statement over the five alternatives in the example.

1. Audio
   Observation

   Even if mobile three is not available in the market, we can go for the next best that is mobile one which is having rank two.

2. Diagram
   Observation

   Mobile 1 has performance score 0.7625 and Rank 2.

If Mobile 3 is unavailable, Mobile 1 is the next best choice because it has rank 2.

1. Ranking is based on the computed performance scores.

Conditional statement about availability of Mobile 3.

1. Audio
   Observation

   The narrator says we can solve or resolve this issue by using a conversion scale, then describes a five point scale.

2. Animation
   Observation

   The overlay Using 5 point Scale appears over the table, and immediately afterward the Looks column changes from words to numbers.

1. Expression

   $$\text{Excellent} \to 5$$
   Explanation

   Mobile 1's Looks entry changes from Excellent to 5.

   Justification

   Five-point scale mapping shown on the slide.

   Shown in the video
2. Expression

   $$\text{Average} \to 3$$
   Explanation

   Mobile 2's Looks entry changes from Average to 3.

   Justification

   Five-point scale mapping shown on the slide.

   Shown in the video
3. Expression

   $$\text{Good} \to 4$$
   Explanation

   Mobile 3 and Mobile 4 change from Good to 4.

   Justification

   Five-point scale mapping shown on the slide.

   Shown in the video
4. Expression

   $$\text{Below Average} \to 2$$
   Explanation

   Mobile 5 changes from Below Average to 2.

   Justification

   Five-point scale mapping shown on the slide.

   Shown in the video

After applying the 5-point scale, the Looks column becomes numerical, allowing the whole table to be treated as a decision matrix.

1. Audio
   Observation

   The narrator says for suppose 300 is the performance value of the third alternative that is the mobile 3 over the first criteria that is the cost.

2. Diagram
   Observation

   The table row Mobile 3 and column Price or Cost contain 300 $. The formula line defines X_{ij} as the performance value of the i-th alternative over the j-th criterion.

1. Expression

   $$i=3,\ j=1$$
   Explanation

   Choose the third alternative and the first criterion.

   Justification

   The narrator identifies Mobile 3 and cost as the example row and column.

   Shown in the video
2. Expression

   $$X_{31}=300$$
   Explanation

   The corresponding cell value is 300.

   Justification

   The table shows 300 $ at the intersection of Mobile 3 and Price or Cost.

   Shown in the video

The entry 300 is the performance value of Mobile 3 with respect to the cost criterion.

1. Audio
   Observation

   Narrator says, Here I'll take the example of storage space. In storage space, 32 GB is the maximum value of the criteria. So we'll divide the performance value of each cell with 32.

2. Animation
   Observation

   Storage Space column changes from 16 GB, 16 GB, 32 GB, 32 GB, 16 GB to $16/32$, $16/32$, $32/32$, $32/32$, $16/32$.

1. Expression

   $$\operatorname{Max}(\text{Storage Space})=32$$
   Explanation

   The maximum storage value in the column is identified as 32 GB.

   Justification

   Directly read from the table and stated in the narration.

   Shown in the video
2. Expression

   $$\frac{16}{32},\ \frac{16}{32},\ \frac{32}{32},\ \frac{32}{32},\ \frac{16}{32}$$
   Explanation

   Each storage entry is divided by 32 to normalize the beneficial criterion.

   Justification

   Application of the beneficial rule $X_{ij}/\operatorname{Max}(X_{ij})$.

   Shown in the video

The Storage Space column is normalized by dividing every entry by the column maximum 32.

1. Audio
   Observation

   Narrator says that for non-beneficial criteria we take the minimum value and divide it with the performance value of each cell.

2. Animation
   Observation

   Price or Cost column changes to $200/250$, $200/200$, $200/300$, $200/275$, $200/225$.

1. Expression

   $$\operatorname{Min}(\text{Price or Cost})=200$$
   Explanation

   The minimum price in the column is identified as 200 dollars.

   Justification

   Directly read from the table and implied by the narration.

   Shown in the video
2. Expression

   $$\frac{200}{250},\ \frac{200}{200},\ \frac{200}{300},\ \frac{200}{275},\ \frac{200}{225}$$
   Explanation

   Each price entry is placed in the denominator under the minimum value 200.

   Justification

   Application of the non-beneficial rule $\operatorname{Min}(X_{ij})/X_{ij}$.

   Shown in the video

The Price or Cost column is normalized by dividing the column minimum 200 by each cell value.

1. Animation
   Observation

   Camera and Looks columns are rewritten as fractions using 16 and 5 as denominators.

2. Animation
   Observation

   All fraction entries are then replaced by decimals and labeled Normalized Decision Matrix.

1. The arithmetic simplification from fractions to decimals is shown as a slide transition rather than written step by step.

1. Expression

   $$\frac{12}{16},\ \frac{8}{16},\ \frac{16}{16},\ \frac{8}{16},\ \frac{16}{16}$$
   Explanation

   The Camera column is normalized using the beneficial rule with maximum 16.

   Justification

   Application of $X_{ij}/\operatorname{Max}(X_{ij})$ to the Camera column.

   Shown in the video
2. Expression

   $$\frac{5}{5},\ \frac{3}{5},\ \frac{4}{5},\ \frac{4}{5},\ \frac{2}{5}$$
   Explanation

   The Looks column is normalized using the beneficial rule with maximum 5.

   Justification

   Application of $X_{ij}/\operatorname{Max}(X_{ij})$ to the Looks column.

   Shown in the video
3. Expression

   $$\begin{bmatrix}0.8&0.5&0.75&1\\1&0.5&0.5&0.6\\0.667&1&1&0.8\\0.727&1&0.5&0.8\\0.889&0.5&1&0.4\end{bmatrix}$$
   Explanation

   The fraction forms are evaluated to produce the displayed normalized decision matrix.

   Justification

   Numerical simplification of the preceding normalized fractions.

   Derived from the video

Applying the two normalization rules to all columns yields the displayed normalized decision matrix.

1. Audio
   Observation

   So we can see here on adding point two, point one two five, point one eight seven five, and point two five, we get a performance score of mobile one as point seven six two five.

2. Diagram
   Observation

   The Mobile 1 row shows $0.2 + 0.125 + 0.1875 + 0.25 = 0.7625$.

1. Expression

   $$0.8 \times 0.25 = 0.2$$
   Explanation

   Multiply the Price or Cost normalized value for Mobile 1 by the decimal weight.

   Justification

   Definition of weighted normalized entry.

   Shown in the video
2. Expression

   $$0.5 \times 0.25 = 0.125$$
   Explanation

   Multiply the Storage Space normalized value for Mobile 1 by the decimal weight.

   Justification

   Definition of weighted normalized entry.

   Shown in the video
3. Expression

   $$0.75 \times 0.25 = 0.1875$$
   Explanation

   Multiply the Camera normalized value for Mobile 1 by the decimal weight.

   Justification

   Definition of weighted normalized entry.

   Shown in the video
4. Expression

   $$1 \times 0.25 = 0.25$$
   Explanation

   Multiply the Looks normalized value for Mobile 1 by the decimal weight.

   Justification

   Definition of weighted normalized entry.

   Shown in the video
5. Expression

   $$0.2 + 0.125 + 0.1875 + 0.25 = 0.7625$$
   Explanation

   Add the four weighted entries to obtain the performance score for Mobile 1.

   Justification

   Definition of performance score as the row sum.

   Shown in the video

The performance score of Mobile 1 is 0.7625.

1. Audio
   Observation

   Similarly, we can obtain the performance score of all other alternative.

2. Diagram
   Observation

   The remaining rows display weighted sums leading to 0.65, 0.8667, 0.7067, and 0.6972. The displayed Mobile 4 total 0.7067 is an arithmetic error; its displayed components total 0.7567.

1. The video does not verbally spell out every intermediate addition for Mobile 2 through Mobile 5, although the arithmetic is visible on screen.

1. Expression

   $$0.25 + 0.125 + 0.125 + 0.15 = 0.65$$
   Explanation

   Row sum for Mobile 2.

   Justification

   Same row-sum method as Mobile 1.

   Shown in the video
2. Expression

   $$0.1667 + 0.25 + 0.25 + 0.2 = 0.8667$$
   Explanation

   Row sum for Mobile 3.

   Justification

   Same row-sum method as Mobile 1.

   Shown in the video
3. Expression

   $$0.1817+0.25+0.125+0.2=0.7567$$
   Explanation

   Corrected Mobile 4 weighted sum; the source slide incorrectly shows 0.7067.

   Justification

   Same row-sum method as Mobile 1.

   Derived from the video
4. Expression

   $$0.2222 + 0.125 + 0.25 + 0.1 = 0.6972$$
   Explanation

   Row sum for Mobile 5.

   Justification

   Same row-sum method as Mobile 1.

   Shown in the video

The corrected scores are Mobile $2 = 0.65$, Mobile $3 = 0.8667$, Mobile $4 = 0.7567$, and Mobile $5 = 0.6972$. The source prints 0.7067 for Mobile 4 by mistake; the order is unchanged.

1. Diagram
   Observation

   The main table lists Mobile 1 through Mobile 5 with columns Price or Cost, Storage Space, Camera, and Looks.

2. Audio
   Observation

   The narrator describes the example as selecting the best mobile phone among all available alternatives based on various criteria like cost, storage space, camera quality, looks.

1. The clip does not complete the actual selection of the best mobile; it only sets up and transforms the data.

Given five mobile alternatives evaluated on four criteria, prepare the data for multi-criteria decision making.

1. Alternatives: Mobile 1, Mobile 2, Mobile 3, Mobile 4, Mobile 5.

2. Criteria: Price or Cost, Storage Space, Camera, Looks.

3. Initial Looks ratings are linguistic: Excellent, Average, Good, Good, Below Average.

Identify the issues in the raw table, convert qualitative ratings to numbers, form a decision matrix, and begin classifying criteria by preference direction.

1. Explanation

   The raw decision table is presented with mixed quantitative units and one qualitative column.

   Justification

   Visible on the slide from the beginning of the clip.

   Shown in the video
2. Expression

   $$\text{Use 5-point scale: } \{\text{Low},\text{Below Average},\text{Average},\text{Good},\text{Excellent}\}\mapsto\{1,2,3,4,5\}$$
   Explanation

   The linguistic column Looks is converted to integers using the displayed scale.

   Justification

   Narration plus the overlay Using 5 point Scale.

   Shown in the video
3. Expression

   $$\text{Looks}:\ 5,3,4,4,2$$
   Explanation

   After substitution, the Looks column becomes numeric for all five alternatives.

   Justification

   Visible after the overlay disappears around 76 seconds.

   Shown in the video
4. Expression

   $$X_{ij}=\text{Performance Value of }i^{th}\text{ Alternative over }j^{th}\text{ criteria}$$
   Explanation

   The transformed table is named a decision matrix, and each cell is interpreted as a performance value.

   Justification

   Bottom label and formula shown on screen; narrator explains the terminology.

   Shown in the video
5. Expression

   $$\text{Price or Cost: Non Beneficial; Storage Space, Camera, Looks: Beneficial}$$
   Explanation

   The criteria are then categorized by whether lower or higher values are preferred.

   Justification

   Orange headers appear above the columns near the end of the clip.

   Shown in the video

The clip prepares the example by converting the qualitative Looks ratings into numbers, defining the resulting table as a decision matrix with entries X_{ij}, and labeling cost as non-beneficial while the other criteria are marked beneficial.

The displayed numeric Looks column matches the 5-point mapping: Excellent→5, Average→3, Good→4, Below Average→2. The example value 300 for Mobile 3 under cost matches the table cell and the narrator's explanation of $X_{31}$.

1. Diagram
   Observation

   Full example table with Mobile 1 to Mobile 5 and four criteria is visible throughout the clip.

2. Audio
   Observation

   Narrator walks through beneficial/non-beneficial classification, normalization, and equal weighting.

1. The final weighted scores and overall ranking are not computed within this clip.

Given five mobile alternatives and four criteria, prepare the data for selecting the best option by normalizing mixed-direction criteria and assigning weights.

1. Alternatives: Mobile 1, Mobile 2, Mobile 3, Mobile 4, Mobile 5.

2. Criteria: Price or Cost, Storage Space, Camera, Looks.

3. Raw values: Mobile $1 = 250$$, 16 GB, 12 MP, 5; Mobile 2 = 200$, 16 GB, 8 MP, 3; Mobile $3 = 300$$, 32 GB, 16 MP, 4; Mobile 4 = 275$, 32 GB, 8 MP, 4; Mobile $5 = 225$$, 16 GB, 16 MP, 2.

4. Criterion types: Price or Cost is non-beneficial; Storage Space, Camera, and Looks are beneficial.

Convert the raw decision table into a normalized decision matrix and then attach criterion weights.

1. Expression

   $$\text{Identify criterion direction}$$
   Explanation

   Classify each column as beneficial or non-beneficial before any arithmetic.

   Justification

   The narrator states that beneficial criteria prefer higher values, while cost prefers the lowest value.

   Shown in the video
2. Expression

   $$\frac{X_{ij}}{\operatorname{Max}(X_{ij})}\ \text{for beneficial columns}$$
   Explanation

   Normalize Storage Space, Camera, and Looks by dividing each entry by its column maximum.

   Justification

   This is the formula shown on screen and explained verbally.

   Shown in the video
3. Expression

   $$\frac{\operatorname{Min}(X_{ij})}{X_{ij}}\ \text{for non-beneficial columns}$$
   Explanation

   Normalize Price or Cost by dividing the column minimum by each entry.

   Justification

   This is the formula shown on screen and explained verbally.

   Shown in the video
4. Expression

   $$\begin{bmatrix}0.8&0.5&0.75&1\\1&0.5&0.5&0.6\\0.667&1&1&0.8\\0.727&1&0.5&0.8\\0.889&0.5&1&0.4\end{bmatrix}$$
   Explanation

   Replace the fraction forms with decimal values to obtain the normalized decision matrix.

   Justification

   The slide displays these decimals after the fraction stage.

   Shown in the video
5. Expression

   $$\text{Weightage}=25\%,\ 25\%,\ 25\%,\ 25\%$$
   Explanation

   Assign equal weights to all four criteria in the next step.

   Justification

   The narrator says equal weightage is allotted to all criteria in this example.

   Shown in the video

The clip ends with a normalized decision matrix and a weightage row of 25% for each criterion; the final weighted ranking is not shown.

Each displayed normalized entry matches the corresponding fraction shown earlier: for example, $16/32=0.5$, $200/250=0.8$, $12/16=0.75$, and $3/5=0.6$.

1. Diagram
   Observation

   A slide titled Selection of the Best shows five alternatives Mobile 1 to Mobile 5 and four criteria Price or Cost, Storage Space, Camera, and Looks.

2. Audio
   Observation

   The narration walks through assigning weights, converting them to decimals, forming the weighted normalized decision matrix, computing performance scores, and ranking the mobiles.

1. The source of the normalized performance values is not explained within this clip.

Choose the best mobile among five alternatives using four criteria and equal weights of 25% each.

1. Alternatives: Mobile 1, Mobile 2, Mobile 3, Mobile 4, Mobile 5.

2. Criteria: Price or Cost, Storage Space, Camera, Looks.

3. Weights: 25% for each criterion, i.e. 0.25 in decimal form.

4. Normalized performance values shown in the initial matrix: Mobile $1 = (0.8, 0.5, 0.75, 1)$; Mobile $2 = (1, 0.5, 0.5, 0.6)$; Mobile $3 = (0.667, 1, 1, 0.8)$; Mobile $4 = (0.727, 1, 0.5, 0.8)$; Mobile $5 = (0.889, 0.5, 1, 0.4)$.

Compute the weighted normalized decision matrix, obtain a performance score for each mobile, and rank the alternatives.

1. Expression

   $$25\% \to 0.25$$
   Explanation

   Convert each percentage weight into decimal form.

   Justification

   Method stated in the video before multiplication.

   Shown in the video
2. Expression

   $$0.8 \times 0.25,\; 0.5 \times 0.25,\; 0.75 \times 0.25,\; 1 \times 0.25$$
   Explanation

   Multiply each normalized performance value by the corresponding criterion weight to form the weighted normalized matrix.

   Justification

   Definition of weighted normalized decision matrix.

   Shown in the video
3. Expression

   $$0.2 + 0.125 + 0.1875 + 0.25 = 0.7625$$
   Explanation

   Add the weighted entries for Mobile 1 to get its performance score.

   Justification

   Definition of performance score as row sum.

   Shown in the video
4. Expression

   $$0.25 + 0.125 + 0.125 + 0.15 = 0.65$$
   Explanation

   Add the weighted entries for Mobile 2.

   Justification

   Same row-sum method.

   Shown in the video
5. Expression

   $$0.1667 + 0.25 + 0.25 + 0.2 = 0.8667$$
   Explanation

   Add the weighted entries for Mobile 3.

   Justification

   Same row-sum method.

   Shown in the video
6. Expression

   $$0.1817+0.25+0.125+0.2=0.7567$$
   Explanation

   Add the weighted entries for Mobile 4.

   Justification

   Same row-sum method.

   Derived from the video
7. Expression

   $$0.2222 + 0.125 + 0.25 + 0.1 = 0.6972$$
   Explanation

   Add the weighted entries for Mobile 5.

   Justification

   Same row-sum method.

   Shown in the video
8. Expression

   $$0.8667>0.7625>0.7567>0.6972>0.65$$
   Explanation

   Order the performance scores from highest to lowest to assign ranks.

   Justification

   Ranking method stated in the video.

   Derived from the video

Corrected scores: Mobile $1 = 0.7625$, Mobile $2 = 0.65$, Mobile $3 = 0.8667$, Mobile $4 = 0.7567$, Mobile $5 = 0.6972$. Ranks: Mobile $3 = 1$, Mobile $1 = 2$, Mobile $4 = 3$, Mobile $5 = 4$, Mobile $2 = 5$. The source slide incorrectly shows 0.7067 for Mobile 4; Mobile 3 remains best.

Independent addition confirms Mobile $4 = 0.7567$; the corrected ordering still matches the rank column shown in the video.

1. Diagram
   Observation

   A static table titled Selection of the Best shows five mobile alternatives and four criteria, with the Looks column containing words rather than numbers.

2. Diagram
   Observation

   A blue text box at the bottom lists SOME ISSUES IN SELECTING THE BEST with two bullets.

1. Title Selection of the Best

2. Table with rows Mobile 1 to Mobile 5

3. Columns Price or Cost, Storage Space, Camera, Looks

4. Blue issue box with two bullets

1. None during this interval; the table remains static while the narrator explains the setup and issues.

1. Five alternatives remain listed throughout.

2. Four criteria remain in the same column order.

3. Looks stays qualitative until the scale overlay appears.

This visual establishes the MCDM problem and makes explicit that the raw data are not yet ready for direct numerical comparison.

1. Animation
   Observation

   An orange overlay titled Using 5 point Scale appears over the middle of the table.

2. Diagram
   Observation

   The overlay lists Low=1, Below Average=2, Average=3, Good=4, Excellent=5.

1. Overlay title Using 5 point Scale

2. Mapping rows Low, Below Average, Average, Good, Excellent

3. Underlying table partially obscured

1. The overlay appears over the table.

2. The mapping from linguistic labels to integers is shown visually.

1. The underlying alternatives and criteria structure do not change.

2. The Looks column is still conceptually the target of conversion.

The animation introduces the rule needed to transform qualitative judgments into numeric matrix entries.

1. Animation
   Observation

   The overlay disappears and the Looks column changes from words to numbers 5, 3, 4, 4, 2.

2. Diagram
   Observation

   The bottom label changes to Decision Matrix with the formula X_{ij} = Performance Value of i^{th} Alternative over j^{th} criteria.

1. Numeric Looks column

2. Bottom label Decision Matrix

3. Formula line for X_{ij}

1. Linguistic entries are replaced by integers.

2. The bottom explanatory box changes from issues to the decision-matrix definition.

1. The same five alternatives and four criteria remain.

2. The other three columns keep their original units and values.

Once all entries are numeric, the table is reinterpreted formally as a decision matrix whose cells are performance values.

1. Animation
   Observation

   Orange headers appear above the columns: Non Beneficial over Price or Cost and Beneficial over Storage Space, Camera, and Looks.

2. Audio
   Observation

   The narrator says the criteria must be categorized into non-beneficial and beneficial criteria and defines non-beneficial as those whose lower value is desired, for example cost.

1. The clip ends before any detailed explanation of how beneficial criteria will be used later.

1. Header Non Beneficial above Price or Cost

2. Headers Beneficial above Storage Space, Camera, Looks

3. Decision matrix below

1. New category labels are added above the criterion columns.

1. The numerical matrix contents remain unchanged.

2. The row order of alternatives remains unchanged.

The visual step encodes preference direction: cost is undesirable when large, whereas the other listed criteria are desirable when large.

1. Diagram
   Observation

   Static table with title Selection of the Best and orange/blue headers.

2. Animation
   Observation

   Mouse cursor points first to Price or Cost, then to Storage Space, Camera, and Looks.

1. Title Selection of the Best

2. Four criterion columns

3. Five mobile rows

4. Mouse cursor

1. Cursor moves across the criterion headers to indicate which attributes are being discussed.

1. The raw numerical table values remain unchanged during this interval.

The visual emphasis introduces the structure of the decision problem and distinguishes the criteria before any transformation.

1. Animation
   Observation

   The word Normalization appears at upper left.

2. Formula
   Observation

   Two formulas appear along the bottom: Non Beneficial- Min(X_ij)/X_ij and Beneficial- X_ij/Max(X_ij).

1. Normalization label

2. Two bottom formulas

3. Original table

1. New text and formulas are added to the slide while the table remains visible.

1. The raw table entries are still unchanged at this moment.

The slide shifts from problem setup to the mathematical rule set needed for comparison.

1. Animation
   Observation

   Storage Space entries become $16/32$, $16/32$, $32/32$, $32/32$, $16/32$.

2. Animation
   Observation

   Price or Cost entries become $200/250$, $200/200$, $200/300$, $200/275$, $200/225$.

3. Animation
   Observation

   Camera and Looks entries later become fractions with denominators 16 and 5.

1. Decision table cells

2. Column maxima and minima

3. Fraction expressions

1. Each column is rewritten from raw units into ratio form according to its criterion type.

1. Row order and column order stay the same.

2. The normalization formulas at the bottom remain visible.

The animation makes the algebraic rule concrete by showing exactly which number goes in the numerator and denominator for each column.

1. Animation
   Observation

   Fraction entries disappear and decimal values fill the table.

2. Diagram
   Observation

   Label Normalized Decision Matrix appears below the table.

1. Decimal table entries

2. Label Normalized Decision Matrix

1. All ratio expressions are replaced by their evaluated decimal values.

1. The table dimensions remain 5 by 4.

This visual change represents completion of the normalization stage.

1. Animation
   Observation

   A new top row labeled Weightage appears.

2. Diagram
   Observation

   Each criterion column receives 25%.

1. Weightage row

2. Four percentage labels

3. Normalized matrix

1. A new header row is inserted above the criteria with 25% repeated four times.

1. The normalized decimal values in the body of the table do not change.

The example advances from normalization to the weighting stage, but stops before computing weighted totals.

1. Diagram
   Observation

   The slide shows a table with Weightage row in orange, Attribute Or Criteria header in blue, Alternative labels on the left, and numeric entries for five mobiles across four criteria.

1. Title Selection of the Best

2. Weightage row

3. Attribute Or Criteria header

4. Alternative column

5. Five mobile rows

6. Four criterion columns

1. The cursor points near the Camera column while the narration discusses higher camera weightage for a selfie lover.

2. At about 14 seconds, 100% appears at the right end of the Weightage row.

1. The normalized performance values in the matrix body remain unchanged during this interval.

2. Each criterion weight remains 25% until the conversion step.

This visual establishes the multi-criteria table before weighting arithmetic begins.

1. Diagram
   Observation

   The Weightage row changes from 25% to 0.25 across all four criteria.

1. Weightage row

1. All four 25% entries are replaced by 0.25.

1. The matrix body values do not change yet.

2. The number of criteria remains four.

The visual prepares the table for direct multiplication by expressing weights as decimal numbers.

1. Diagram
   Observation

   Every cell in the matrix body is rewritten as a product of the original value and 0.25, for example 0.8 x 0.25 and 1 x 0.25.

1. Matrix body cells

1. Each numeric entry is replaced by an explicit multiplication expression with 0.25.

1. The row and column structure stays the same.

2. The weights remain 0.25 for all criteria.

This makes the weighting operation visible before the products are simplified.

1. Audio
   Observation

   The narrator explicitly says we cannot compare cost with storage space or camera quality because the units differ.

2. Diagram
   Observation

   The bullet All Criteria have different Unit is displayed.

One might think the entries in different criterion columns can be compared directly because they are all numbers in the same table.

The clip stresses that dollars, gigabytes, and megapixels measure different things, so raw cross-criterion comparison is invalid before normalization or other processing.

1. Audio
   Observation

   The narrator identifies use of linguistic term as an issue and then says it can be resolved by using a conversion scale.

2. Diagram
   Observation

   The Looks column initially contains words and is later replaced by integers via the 5-point scale.

Words such as Excellent or Average may seem sufficient as final decision-matrix entries.

In this presentation, qualitative labels must first be mapped to numbers; only then is the table called a decision matrix with performance values.

1. Audio
   Observation

   Narrator says there is a difference in both beneficial and non-beneficial: maximum value takes the denominator and in non-beneficial the minimum value takes the numerator.

One might apply the same fraction pattern to all columns regardless of whether larger or smaller raw values are better.

The video explicitly contrasts the two cases: beneficial criteria use $X_{ij}/\operatorname{Max}(X_{ij})$, while non-beneficial criteria use $\operatorname{Min}(X_{ij})/X_{ij}$.

1. Audio
   Observation

   For some selfie lover, camera will have a higher weightage. They may assign the camera a weightage of around 50 to 60 percent.

One might assume every decision-maker must use the same criterion weights shown in a given example.

The video explicitly says weights can differ by person; a selfie lover may give camera a much larger share, around 50% to 60%.

1. Audio
   Observation

   One important thing to note here is the sum of weightage is always 100 or 100 percent.

One might assign arbitrary percentages to criteria without checking that they total 100%.

The video stresses that the sum of all weightages must be 100% before proceeding.

1. Audio
   Observation

   Convert the percent into number and then multiply.

2. Diagram
   Observation

   The table changes from 25% to 0.25 before any multiplication is shown.

One might try to multiply matrix entries by 25% without first rewriting the weight as a decimal number.

The video shows the required conversion step: 25% becomes 0.25 before forming products such as 0.8 x 0.25.

1. Audio
   Observation

   The narrator first introduces the MCDM mobile-selection example and then immediately lists issues in selecting the best.

2. Diagram
   Observation

   The same table remains on screen while the issue box is introduced below it.

The general MCDM example contains the specific difficulty that its criteria are measured in different units.

1. Audio
   Observation

   The narrator says the linguistic-term issue can be solved by using a conversion scale.

2. Animation
   Observation

   The 5-point scale overlay appears directly in response to the qualitative Looks column.

The five-point scale is applied specifically to resolve the problem of linguistic terms in the Looks criterion.

1. Animation
   Observation

   After the scale is shown, the Looks column becomes numeric and the bottom label changes to Decision Matrix.

2. Audio
   Observation

   The narrator says after substituting the numerical value we get a decision matrix.

Converting qualitative labels into numbers is a prerequisite for treating the whole table as a decision matrix with entries X_{ij}.

1. Audio
   Observation

   The narrator introduces the decision matrix and then says next we have to categorize the criteria into non-beneficial and beneficial criteria.

2. Animation
   Observation

   Beneficial / Non Beneficial headers are added above the already numeric matrix.

Once the table is expressed numerically as a decision matrix, the next step is to assign preference directions to the criteria.

1. Audio
   Observation

   The narrator gives cost as the example of a non-beneficial criterion.

2. Diagram
   Observation

   The header Non Beneficial is placed above Price or Cost.

Within the mobile-selection example, Price or Cost is the special case illustrating a non-beneficial criterion.

1. Diagram
   Observation

   The initial table is annotated with Non Beneficial and Beneficial headers.

The decision table contains the criterion classification needed before normalization.

1. Audio
   Observation

   Narrator links comparability of all criteria to performing normalization after discussing beneficial and non-beneficial attributes.

Knowing which criteria are beneficial or non-beneficial is required before choosing the correct normalization rule.

1. Formula
   Observation

   The beneficial formula is presented as one of the two normalization rules.

The general purpose of normalization is implemented for beneficial columns by the formula $X_{ij}/\operatorname{Max}(X_{ij})$.

1. Formula
   Observation

   The non-beneficial formula is presented as the other normalization rule.

The general purpose of normalization is implemented for non-beneficial columns by the formula $\operatorname{Min}(X_{ij})/X_{ij}$.

1. Animation
   Observation

   After all columns are converted, the result is labeled Normalized Decision Matrix.

The normalized decision matrix depends on applying the beneficial rule to the beneficial columns.

1. Animation
   Observation

   The Price or Cost column is included in the final decimal matrix.

The normalized decision matrix also depends on applying the non-beneficial rule to the cost column.

1. Audio
   Observation

   Narrator says the next step is to assign the weightage to the criteria after obtaining the normalized decision matrix.

Weight assignment is presented as the step that follows normalization.

1. Audio
   Observation

   The narrator introduces MCDM and the mobile-phone selection example.

1. MCDM example setup: selecting the best mobile phone

1. Audio
   Observation

   The narrator explains that different units prevent direct comparison among cost, storage space, and camera quality.

1. Issue 1: criteria measured in different units
2. Mistaking raw heterogeneous criterion values for directly comparable numbers

1. Audio
   Observation

   The narrator discusses linguistic terms and then introduces a conversion scale.

2. Diagram
   Observation

   The 5-point scale overlay maps words to numbers.

1. Issue 2: qualitative linguistic ratings
2. Five-point conversion scale for linguistic terms

1. Formula
   Observation

   The slide displays X_{ij} = Performance Value of i^{th} Alternative over j^{th} criteria.

2. Audio
   Observation

   The narrator defines each cell as a performance value and gives the example $X_{31}=300$.

1. Decision matrix and performance value
2. X_{ij}
3. Example identifying a specific matrix entry

1. Audio
   Observation

   The narrator defines non-beneficial criteria as those whose lower value is desired, for example cost.

2. Diagram
   Observation

   Non Beneficial is placed above Price or Cost.

1. Classification of criteria as beneficial or non-beneficial
2. Definition of non-beneficial criteria

1. Diagram
   Observation

   Beneficial headers appear above Storage Space, Camera, and Looks.

1. The clip labels these columns as Beneficial but does not yet explain the full downstream method using that label.

1. Classification of criteria as beneficial or non-beneficial

1. Audio
   Observation

   Narrator defines beneficial criteria as those whose higher value is desired.

1. Beneficial versus non-beneficial criteria

1. Audio
   Observation

   Narrator says normalization is done to make all criteria comparable.

1. Purpose of normalization

1. Formula
   Observation

   Non-beneficial formula shown on screen.

2. Animation
   Observation

   Price or Cost column rewritten as 200 divided by each cell value.

1. Non-beneficial normalization formula
2. Worked normalization of the Price or Cost column

1. Audio
   Observation

   Narrator uses storage space as the example for beneficial normalization.

2. Animation
   Observation

   Storage Space column rewritten with denominator 32.

1. Beneficial normalization formula
2. Worked normalization of the Storage Space column

1. Audio
   Observation

   Narrator explicitly contrasts denominator use in beneficial criteria with numerator use in non-beneficial criteria.

1. Mixing up numerator and denominator across criterion types
2. Beneficial normalization formula
3. Non-beneficial normalization formula

1. Diagram
   Observation

   Weightage row appears after the normalized matrix.

2. Audio
   Observation

   Narrator says the next step is to assign the weightage to the criteria.

1. Assigning criterion weights