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

Civic Mazda 6 7 8 8.4 7.6 7.5 7.0 Style Reliability Fuel Eco. Saturn Ford 7 9 9 8 7 8 9 6 8

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

Civic Mazda 6 7 8 6 Cost Style Reliability Fuel Eco. Saturn Ford 7 9 9 8 8 7 8 7 9 6 8 9

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

Style Rel. Fuel Saturn Ford 49 81 81 64 64 49 64 49 81 36 64 81 Civic Mazda Cost x ij 2 i ( x 2 ) 1/2 36 49 64 36 230 215 273 230 15.17 14.66 16.52 15.17

Steps of TOPSIS

• Step 1 (b): divide each column by ( x 2 ij ) 1/2 to get r ij

Style Rel. Fuel Saturn Ford 0.46 0.61 0.54 0.53 0.53 0.48 0.48 0.46 0.59 0.41 0.48 0.59 Civic Mazda 0.40 0.48 0.48 0.40 Cost

Steps of TOPSIS

• Step 2 (b): multiply each column by w j to get v ij .

Style Rel. Fuel Saturn Ford 0.046 0.244 0.162 0.106 0.053 0.192 0.144 0.092 0.059 0.164 0.144 0.118 Civic Mazda 0.040 0.192 0.144 0.080 Cost

Steps of TOPSIS

• Step 3 (a): determine ideal solution A*.

• A* = {0.059, 0.244, 0.162, 0.080}

Style Rel. Fuel Saturn Ford 0.046 0.244 0.162 0.106 0.053 0.192 0.144 0.092 0.059 0.164 0.144 0.118 Civic Mazda 0.040 0.192 0.144 0.080 Cost

Steps of TOPSIS

• Step 3 (a): find negative ideal solution A ' .

• A ' = {0.040, 0.164, 0.144, 0.118}

Style Rel. Fuel Saturn Ford 0.046 0.244 0.162 0.106 0.053 0.192 0.144 0.092 0.059 0.164 0.144 0.118 Civic Mazda 0.040 0.192 0.144 0.080 Cost

Steps of TOPSIS

• Step 4 (a): determine separation from ideal solution A* = {0.059, 0.244, 0.162, 0.080} S i * = [ (v j * v ij ) 2 ] for each row j

Style Rel. Fuel Saturn Ford (.046 -.059 ) 2 (.244 -.244 ) 2 (0) 2 (.026) 2 Civic Mazda Cost (.053 -.059 ) 2 (.192 -.244 ) 2 (-.018) 2 (.012) 2 (.053 -.059 ) 2 (.164 -.244 ) 2 (-.018) 2 (.038) 2 (.053 -.059 ) 2 (.192 -.244 ) 2 (-.018) 2 (.0) 2

Steps of TOPSIS

• Step 4 (a): determine separation from ideal solution S i *

(v j * v ij ) 2 S i * = [ (v j * v ij ) 2 ] Saturn Ford 0.000845 0.029 0.003208 0.057 0.008186 0.090 Civic Mazda 0.003389 0.058

Steps of TOPSIS

• Step 4 (b): find separation from negative ideal solution A ' = {0.040, 0.164, 0.144, 0.118}

• S i ' = [ (v j ' v ij ) 2 ] for each row j

Style Rel. Fuel Saturn Ford (.046 -.040 ) 2 (.244 -.164 ) 2 (.018) 2 (-.012) 2 Civic Mazda Cost (.053 -.040 ) 2 (.192 -.164 ) 2 (0) 2 (-.026) 2 (.053 -.040 ) 2 (.164 -.164 ) 2 (0) 2 (0) 2 (.053 -.040 ) 2 (.192 -.164 ) 2 (0) 2 (-.038) 2

Steps of TOPSIS

• Step 4 (b): determine separation from negative ideal solution S i '

(v j ' v ij ) 2 S i ' = [ (v j ' v ij ) 2 ] Saturn Ford 0.006904 0.083 0.001629 0.040 0.000361 0.019 Civic Mazda 0.002228 0.047

Steps of TOPSIS

• Step 5: Calculate the relative closeness to the ideal solution C i * = S ' i / (S i * +S ' i )

S ' i /(S i * +S ' i ) C i * Saturn Ford 0.083/0.112 0.74 BEST 0.040/0.097 0.41 0.019/0.109 0.17 Civic Mazda 0.047/0.105 0.45