# weighted score and topsis

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- Multi-Criteria Decision Making MCDM Approaches
- Introduction
- Zeleny (1982) opens his book Multiple Criteria Decision Making with a statement:

- It has become more and more difficult to see the world around us in a unidimensional way and to use only a single criterion when judging what we see

- Introduction
- Many public sector problems and even private decision involve multiple objectives and goals. As an example:

- Locating a nuclear power plant involves objectives such as:

- Safety

- Health

- Environment

- Cost

- Examples of Multi-Criteria Problems
- In a case study on the management of R&D research (Moore et. al 1976) , the following objectives have been identified:

- Profitability

- Growth and diversity of the product line

- Increased market share

- Maintained technical capability

- Firm reputation and image

- Research that anticipates competition

- Examples of Multi-Criteria Problems
- In determining an electric route for power transmission in a city, several objectives could be considered:

- Cost

- Health

- Reliability

- Importance of areas

- Examples of Multi-Criteria Problems
- In selecting a major at KFUPM, several objectives can be considered. These objectives or criteria include:

- Job market upon graduation

- Job pay and opportunity to progress

- Interest in the major

- Likelihood of success in the major

- Future job image

- Parent wish

- Examples of Multi-Criteria Problems
- Wife selection problem . This problem is a good example of multi-criteria decision problem. Criteria include:

- Religion

- Beauty

- Wealth

- Family status

- Family relationship

- Education

- Approaches For MCDM
- Several approaches for MCDM exist. We will cover the following:

- Weighted score method ( Section 5.1 in text book).

- TOPSIS method

- Analytic Hierarchy Process (AHP)

- Goal programming ?

- Weighted score method
- Determine the criteria for the problem

- Determine the weight for each criteria. The weight can be obtained via survey, AHP, etc.

- Obtain the score of option i using each criteria j for all i and j

- Compute the sum of the weighted score for each option .

- Weighted score method
- In order for the sum to make sense all criteria scale must be consistent, i.e.,

- More is better or less is better for all criteria

- Example:

- In the wife selection problem , all criteria (Religion, Beauty, Wealth, Family status, Family relationship, Education) more is better

- If we consider other criteria (age, dowry) less is better

- Weighted score method
- Let S ij score of option i using criterion j

- w j weight for criterion j

- S i score of option i is given as:

- S i = w j S ij

- j

- The option with the best score is selected.

- Weighted Score Method
- The method can be modified by using U(S ij ) and then calculating the weighted utility score.

- To use utility the condition of separability must hold.

- Explain the meaning of separability:

- U(S i ) = w j U(S ij )

- U(S i ) U( w j S ij )

- Example Using Weighted Scoring Method
- Objective

- Selecting a car

- Criteria

- Style, Reliability, Fuel-economy

- Alternatives

- Civic Coupe, Saturn Coupe, Ford Escort, Mazda Miata

- Weights and Scores
- Weight 0.3 0.4 0.3 S i

- TOPSIS METHOD
- T echnique of O rder P reference by S imilarity to I deal S olution

- This method considers three types of attributes or criteria

- Qualitative benefit attributes/criteria

- Quantitative benefit attributes

- Cost attributes or criteria

- TOPSIS METHOD
- In this method two artificial alternatives are hypothesized :

- Ideal alternative : the one which has the best level for all attributes considered.

- Negative ideal alternative : the one which has the worst attribute values.

- TOPSIS selects the alternative that is the closest to the ideal solution and farthest from negative ideal alternative.

- Input to TOPSIS
- TOPSIS assumes that we have m alternatives (options) and n attributes/criteria and we have the score of each option with respect to each criterion.

- Let x ij score of option i with respect to criterion j

- We have a matrix X = (x ij ) m n matrix.

- Let J be the set of benefit attributes or criteria (more is better)

- Let J ' be the set of negative attributes or criteria (less is better)

- Steps of TOPSIS
- Step 1: Construct normalized decision matrix.

- This step transforms various attribute dimensions into non-dimensional attributes, which allows comparisons across criteria.

- Normalize scores or data as follows:

- r ij = x ij / ( x 2 ij ) for i = 1, , m; j = 1, , n

- i

- Steps of TOPSIS
- Step 2: Construct the weighted normalized decision matrix.

- Assume we have a set of weights for each criteria w j for j = 1,n.

- Multiply each column of the normalized decision matrix by its associated weight.

- An element of the new matrix is:

- v ij = w j r ij

- Steps of TOPSIS
- Step 3: Determine the ideal and negative ideal solutions.

- Ideal solution.

- A* = { v 1 * , , v n * }, where

- v j * ={ max (v ij ) if j J ; min (v ij ) if j J ' }

- i i

- Negative ideal solution.

- A' = { v 1 ' , , v n ' }, where

- v' = { min (v ij ) if j J ; max (v ij ) if j J ' }

- i i

- Steps of TOPSIS
- Step 4: Calculate the separation measures for each alternative.

- The separation from the ideal alternative is:

- S i * = [ (v j * v ij ) 2 ] i = 1, , m

- j

- Similarly, the separation from the negative ideal alternative is:

- S ' i = [ ( v j ' v ij ) 2 ] i = 1, , m

- j

- Steps of TOPSIS
- Step 5: Calculate the relative closeness to the ideal solution C i *

- C i * = S ' i / (S i * +S ' i ) , 0 C i * 1

- Select the option with C i * closest to 1.

- WHY ?

- Applying TOPSIS Method to Example
- Weight 0.1 0.4 0.3 0.2

- Applying TOPSIS to Example
- m = 4 alternatives (car models)

- n = 4 attributes/criteria

- x ij = score of option i with respect to criterion j

- X = {x ij } 4 4 score matrix.

- J = set of benefit attributes: style, reliability, fuel economy (more is better)

- J ' = set of negative attributes: cost (less is better)

- Steps of TOPSIS
- Step 1(a): calculate ( x 2 ij ) 1/2 for each column

- Steps of TOPSIS
- Step 1 (b): divide each column by ( x 2 ij ) 1/2 to get r ij

- Steps of TOPSIS
- Step 2 (b): multiply each column by w j to get v ij .

- Steps of TOPSIS
- Step 3 (a): determine ideal solution A*.

- A* = {0.059, 0.244, 0.162, 0.080}