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Multi-Criteria Decision Making

MCDM Approaches

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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”

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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

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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

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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

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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

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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

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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 ?

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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 .

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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

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Weighted score method

Let Sij score of option i using criterion j

wj weight for criterion j

Si score of option i is given as:

Si = wj Sij

j

The option with the best score is selected.

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Weighted Score Method

The method can be modified by using U(Sij) and then calculating the weighted utility score.

To use utility the condition of separability must hold.

Explain the meaning of separability:

U(Si) = wj U(Sij)

U(Si) U( wj Sij)

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Example Using Weighted Scoring Method

Objective• Selecting a car

Criteria• Style, Reliability, Fuel-economy

Alternatives• Civic Coupe, Saturn Coupe, Ford Escort,

Mazda Miata

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Weights and Scores

Weight 0.3 0.4 0.3 Si

Style Reliability Fuel Eco.

Saturn

Ford

7 9 9

8 7 8

9 6 8

Civic

Mazda

6 7 8

8.4

7.6

7.5

7.0

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TOPSIS METHOD

Technique of Order Preference by Similarity to Ideal Solution

This method considers three types of attributes or criteria

• Qualitative benefit attributes/criteria• Quantitative benefit attributes• Cost attributes or criteria

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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.

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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 xij score of option i with respect to criterion j

We have a matrix X = (xij) mn 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)

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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:

rij = xij/ (x2ij) for i = 1, …, m; j = 1, …, n

i

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Steps of TOPSIS

Step 2: Construct the weighted normalized decision matrix.

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

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

An element of the new matrix is:

vij = wj rij

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Steps of TOPSIS Step 3: Determine the ideal and negative ideal

solutions.

Ideal solution.

A* = { v1*

, …, vn*}, where

vj*

={ max (vij) if j J ; min (vij) if j J' } i i

Negative ideal solution.

A' = { v1' , …, vn' }, where

v' = { min (vij) if j J ; max (vij) if j J' } i i

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Steps of TOPSIS

Step 4: Calculate the separation measures for each alternative.

The separation from the ideal alternative is:

Si *

= [ (vj*– vij)2 ] ½ i = 1, …, m

j

Similarly, the separation from the negative ideal alternative is:

S'i = [ (vj' – vij)2 ] ½ i = 1, …, m j

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Steps of TOPSIS

Step 5: Calculate the relative closeness to the ideal solution Ci

*

Ci* = S'i / (Si

* +S'i ) , 0 Ci* 1

Select the option with Ci* closest to 1.

WHY ?

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Applying TOPSIS Method to Example

Weight 0.1 0.4 0.3 0.2

Style Reliability Fuel Eco.

Saturn

Ford

7 9 9 8

8 7 8 7

9 6 8 9

Civic

Mazda

6 7 8 6

Cost

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Applying TOPSIS to Example

m = 4 alternatives (car models) n = 4 attributes/criteria

xij = score of option i with respect to criterion j

X = {xij} 44 score matrix.

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

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

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Steps of TOPSIS

Step 1(a): calculate (x2ij )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

xij2i

(x2)1/2

36 49 64 36

230 215 273 230

15.17 14.66 16.52 15.17

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Steps of TOPSIS

Step 1 (b): divide each column by (x2ij )1/2

to get rij

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

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Steps of TOPSIS

Step 2 (b): multiply each column by wj to get vij.

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

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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

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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

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Steps of TOPSIS

Step 4 (a): determine separation from ideal solution A* = {0.059, 0.244, 0.162, 0.080} Si

* = [ (vj

*– vij)2 ] ½ 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

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Steps of TOPSIS

Step 4 (a): determine separation from ideal solution Si

*

(vj

*–vij)2 Si* = [ (vj

*– vij)2 ] ½

Saturn

Ford

0.000845 0.029

0.003208 0.057

0.008186 0.090

Civic

Mazda 0.003389 0.058

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Steps of TOPSIS

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

Si' = [ (vj'– vij)2 ] ½ for each row j

Style Rel. Fuel

Saturn

Ford

(.046-.040)2 (.244-.164)2 (.018)2 (-.012)2Civic

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

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Steps of TOPSIS

Step 4 (b): determine separation from negative ideal solution Si'

(vj'–vij)2 Si' = [ (vj'– vij)2 ] ½

Saturn

Ford

0.006904 0.083

0.001629 0.040

0.000361 0.019

Civic

Mazda 0.002228 0.047

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Steps of TOPSIS

Step 5: Calculate the relative closeness to the ideal solution Ci

* = S'i / (Si

* +S'i )

S'i /(Si

*+S'i) Ci*

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