MADM

Y. İlker TOPCU, Ph.D.

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Decision making may be defined as: • Intentional and reflective choice in response to

perceived needs (Kleindorfer et al., 1993)

• Decision maker’s (DM’s) choice of one alternative or a subset of alternatives among all possible alternatives with respect to her/his goal or goals (Evren and Ülengin, 1992)

• Solving a problem by choosing, ranking, or classifying over the available alternatives that are characterized by multiple criteria (Topcu, 1999)

Multicriteria Decision Making

Multicriteria Decision Making

• A single DM is to choose among a countable (usually finite) or uncountable set of alternatives that s/he evaluates on the basis of two or more (multiple) criteria (Korhonen et al., 1992; Dyer et al., 1992)

• MCDM consists of constructing a global preference relation for a set of alternatives evaluated using several criteria (Vansnick, 1986)

• The aim of any MCDM technique is to provide help and guidance to the DM in discovering his or her most desired solution to the problem (Stewart, 1992)

MADM – MODM

A differentiation can be made w.r.t. number of alternatives:

• Multi Attribute Decision Making – MADM) Cases in which the set of decision alternatives is defined explicitly by a finite list of alternative actions – Discrete alternatives

• Multi Objective Decision Making – MODM) Those in which a is defined implicitly by a mathematical programming structure – Continuous alternatives

Multi Attribute Decision Making

• MADM is making preference decisions (selecting, ranking, screening, prioritization, classification) over the available alternatives (finite number) that are characterized by attributes (multiple, conflicting, weighted, and incommensurable) (Yoon & Hwang, 1995)

MADM Problem Statements

Problematiques:• Choice (a)• Classification/Sorting (b)• Ranking (g)

Choice

• Isolate the smallest subset liable to justify the elimination of all other actions

• Selecting a subset, as restricted as possible, containing the most satisfactory alternatives as a compromise solution

Classification

• Sorting alternatives and assigning each of them into prespecified / predefined categories

Ranking

• Building a partial or complete pre-order as rich as possible

• Ranking (all or some of) alternatives by decreasing order of preference

Decision Making Process

1. Structuring the ProblemExploring the issue and determining whether or not MADM is an appropriate tool: If so, then alternatives for evaluation and relevant criteria can be expected to emerge

2. Constructing the Decision ModelElicitation of preferences, performance values, and (if necessary) weights

3. Analyzing (Solving) the ProblemUsing a solution method to synthesize and explore results (through sensitivity and robustness analyses)

• Alternative evaluations w.r.t. attributes are presented in a decision matrix• Entries are performance values • Rows represent alternatives• Columns represent attributes

Decision Matrix

Attributes

• Benefit attributesOffer increasing monotonic utility. Greater the attribute value the more its preference

• Cost attributesOffer decreasing monotonic utility. Greater the attribute value the less its preference

• Nonmonotonic attributesOffer nonmonotonic utility. The maximum utility is located somewhere in the middle of an attribute range

Global Performance Value

• If solution method that will be utilized is performance aggregation oriented, performance values should be aggregated.

• In this case• Performance values are normalized to eliminate

computational problems caused by differing measurement units in a decision matrix

• Relative importance of attributes are determined

Normalization

• Aims at obtaining comparable scales, which allow interattribute as well as intra-attribute comparisons

• Normalized performance values have dimensionless units

• The larger the normalized value becomes, the more preference it has

Normalization Methods

1. Distance-Based Normalization Methods

2. Proportion Based Normalization Methods (Standardization)

Distance-Based Normalization Methods

If we define the normalized rating as the ratio between individual and combined distance from the origin (0,0,…,0) then the comparable rating of xij is

given as (Yoon and Kim, 1989):

rij(p) = (xij - 0) /

This equation is arranged for benefit attributes.

Cost attributes become benefit attributes by taking the inverse rating (1/ xij)

ppm

kkjx

/1

1

0

Distance-Based Normalization Methods

• Normalization (p=1: Manhattan distance)• Vector Normalization (p=2: Euclidean distance)• Linear Normalization (p= : Tchebycheff dist.)

rij(1) = xij /

rij(2) = xij /

rij( ) = xij / maks (BENEFIT ATTRIBUTE)

rij( ) = min / xij (COST ATTRIBUTE)

m

kkjx

1

2

1

m

kkjx

,...,2,1 , mkxkj

,...,2,1 , mkxkj

Proporiton-Based Normalization Methods

The proportion of difference between performance value of the alternative and the worst performance value to difference between the best and the worst performance values (Bana E Costa, 1988; Kirkwood, 1997)

rij = (xij – xj-) / (xj

* – xj-) benefit attribute

rij = (xj- – xij) / (xj

- – xj*) cost attribute

where * represents the best and – represents the worst

(best: max. perf. value for benefit; min. perf. value for cost or ideal value determined by DM for that attribute)

Example

Transformation of Nonmonotonic Attributes to Monotonic

• Statistical z score is taken: exp(–z2/2)

where

z = (xij – xj0) / sj

xj0 is the most favorable performance value w.r.t..attribute j.

sj is the standard deviation of performance values w.r.t. attribute j.

Example

Decision Matrix for “Buying a New Car” Problem

Price Comfort Perf. DesignWeight 5 4 3 3

a 1 300 E (3) E (3) S (3)

a 2 250 E (3) A (2) S (3)

a 3 250 A (2) E (3) S (3)

a 4 200 A (2) E (3) O (2)

a 5 200 A (2) A (2) S (3)

a 6 200 W (1) E (3) S (3)

a 7 100 W (1) A (2) O (2)

E:Excellent; A:Average, W:Weak, S:Superior, O:Ordinary

SAW

• Simple Additive Weighting – Weighted Average – Weighted Sum (Yoon & Hwang, 1995; Vincke, 1992...)

• A global (total) score in the SAW is obtained by adding contributions from each attribute.

• A common numerical scaling system such as normalization (instead of single dimensional value functions) is required to permit addition among attribute values.

• Value (global score) of an alternative can be expressed as:

V(ai) = Vi =

n

jijj rw

1

Example for SAW

Normalized (Linear) Decision Matrix and Global Scores

Price Comfort Perf. DesignNorm. w 0,3333 0,2667 0,2 0,2a 1 0,3333 1 1 1 0,7778

a 2 0,4 1 0,6667 1 0,7334

a 3 0,4 0,6667 1 1 0,7111

a 4 0,5 0,6667 1 0,6667 0,6778

a 5 0,5 0,6667 0,6667 1 0,6778

a 6 0,5 0,3333 1 1 0,6555

a 7 1 0,3333 0,6667 0,6667 0,6889

V,

WP

• Weighted Product (Yoon & Hwang, 1995)

• Normalization is not necessary!• When WP is used weights become exponents

associated with each attribute value;• a positive power for benefit attributes• a negative power for cost attributes

• Because of the exponent property, this method requires that all ratings be greater than 1. When an attribute has fractional ratings, all ratings in that attribute are multiplied by 10m to meet this requirement

Vi = j

wij

jx )(

Example for WP

Quantitative Decision Matrix and Global Scores

Price Comfort Perf. DesignNorm. w 0,3333 0,2667 0,2 0,2a 1 300 3 3 3 0,3108

a 2 250 3 2 3 0,3045

a 3 250 2 3 3 0,2964

a 4 200 2 3 2 0,2944

a 5 200 2 2 3 0,2944

a 6 200 1 3 3 0,2654

a 7 100 1 2 2 0,2843

v(ai)

TOPSIS

• Technique for Order Preference by Similarity to Ideal Solution (Yoon & Hwang, 1995; Hwang & Lin, 1987)

• Concept:Chosen alternative should have the shortest distance from the positive ideal solution and the longest distance from the negative ideal solution

• Steps:• Calculate normalized ratings• Calculate weighted normalized ratings• Identify positive-ideal and negative-ideal solutions• Calculate separation measures• Calculate similarities to positive-ideal solution• Rank preference order

Steps

• Calculate normalized ratings• Vector normalization (Euclidean) is used• Do not take the inverse rating for cost attributes!

• Calculate weighted normalized ratings• vij = wj * rij

• Identify positive-ideal and negative-ideal solutions

where J1 is a set of benefit attributes and J2 is a set of cost attributes

*a },...,,...,,{ ***2

*1 nj vvvv

miJjvJjv ij

iij

i,...,1 min,maks 21= =

= =a },...,,...,,{ 21

nj vvvv

miJjvJjv ij

iij

i,...,1 maks,min 21

Steps

• Calculate separation measures• Euclidean distance (separation) of each alternative from the

ideal solutions are measured:

• Calculate similarities to positive-ideal solution

• Rank preference order• Rank the alternatives according to similarities in descending

order.• Recommend the alternative with the maximum similarity

j

jiji vvS 2** )( j

jiji vvS 2)(

)/( ** iiii SSSC

Example for TOPSIS

• Normalized (Vector) Decision Matrix

Price Comfort Perf. DesignNorm. w 0,3333 0,2667 0,2 0,2

a 1 0,5108 0,5303 0,433 0,4121

a 2 0,4256 0,5303 0,2887 0,4121

a 3 0,4256 0,3536 0,433 0,4121

a 4 0,3405 0,3536 0,433 0,2747

a 5 0,3405 0,3536 0,2887 0,4121

a 6 0,3405 0,1768 0,433 0,4121

a 7 0,1703 0,1768 0,2887 0,2747

Weighted Normalized Ratings &Positive–Negative Ideal

Price Comfort Perf. Design

a 1 0,1703 0,1414 0,0866 0,0824

a 2 0,1419 0,1414 0,0577 0,0824

a 3 0,1419 0,0943 0,0866 0,0824

a 4 0,1135 0,0943 0,0866 0,0549

a 5 0,1135 0,0943 0,0577 0,0824

a 6 0,1135 0,0471 0,0866 0,0824

a 7 0,0568 0,0471 0,0577 0,0549

a* 0,0568 0,1414 0,0866 0,0824a- 0,1703 0,0471 0,0577 0,0549

Separation Measures & Similarities to Positive Ideal Solution

Rank

a 1 0.1135 0.1024 0.4742 5

a 2 0.0899 0.1022 0.5321 1

a 3 0.0973 0.0679 0.4111 6

a 4 0.0787 0.0792 0.5016 3

a 5 0.0792 0.0787 0.4984 4

a 6 0.11 0.0693 0.3866 7

a 7 0.1024 0.1135 0.5258 2

*S S *C