Constructing the Decision
Model
Y. İlker TOPCU, Ph.D.
www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info
www.facebook.com/yitopcu
twitter.com/yitopcu
Decision Matrix
• Alternative evaluations w.r.t. attributes are presented
in a decision matrix
• Entries are performance values
• Rows represent alternatives
• Columns represent attributes
Example
Car purchasing decision
Selling price Fuel cons.
Comfort
K TL (cost)
lt/100km (cost)
1-5 (benefit)
a
40
10
5
b
30
9
4
c
35
8
4
Example
Material Selection Problem
•
Consider a material selection problem for a complex 3D part which needs to be
designed for operation in a high-temperature oxygen-rich environment, under
extreme bending fatigue loads. The design is not intended for mass production.
The decision maker (analyst) considers 6 alternative materials and 4 factors with
the data shown below. The quantification of the corrosion resistance data has
been previously done using an 11-point scale.
Material selection factors
Machinability
Material Hardness rating(%) Cost($/lb) Corrosion resitance
1
420
25
5
Extremely high (0.865)
2
350
40
3
High (0.665)
3
390
30
3
High (0.665)
4
250
35
1.3
High (0.665)
5
600
30
2.2
Very high (0.745)
6
230
55
4
Average (0.5)
Attributes
• Benefit attributes
Offer increasing monotonic utility. Greater the
attribute value the more its preference
• Cost attributes
Offer decreasing monotonic utility. Greater the
attribute value the less its preference
• Nonmonotonic attributes
Offer 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) /  xkj  0

 k 1
m
p
1/ p





This equation is arranged for benefit attributes.
Cost attributes become benefit attributes by taking
the inverse rating (1/ xij)
Distance-Based Normalization Methods
• Normalization
(p=1: Manhattan distance)
• Vector Normalization (p=2: Euclidean distance)
• Linear Normalization (p=  : Tchebycheff dist.)
m
rij(1) = xij /
x
k 1
kj
2
m
x
rij(2) = xij /
k 1
kj

rij( ) = xij / max xkj , k  1,2,...,m



rij() = min xkj , k  1,2,...,m / xij
(BENEFIT ATTRIBUTE)
(COST ATTRIBUTE)
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
• exp(–z2/2) exponential function is utilized for
transformation
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
Attribute Weighting
• Most methods translate the relative importance of
attributes into numbers which are often called as
“weights” (Vincke, 1992)
• Methods utilized for assignment of weights can be
classified in two groups (Huylenbroeck, 1995; Munda 1993; AlKloub et al., 1997; Kleindorfer et al., 1993; Yoon and Hwang, 1995):
• Direct Determination
• Indirect Determination
Weight Assignment Methods
• Direct Determination
• Rating, Point allocation, Categorization
• Ranking
• Swing
• Trade-off
• Ratio (Eigenvector prioritization)
• Indirect Determination
• Centrality
• Regression – Conjoint analysis
• Interactive
Swing method
• Given is a set of alternatives and a set of attributes. Let N be the
number of attributes.
1. Determine the best and worst value of each attribute over the set of
alternatives.
2. Create N+1 fictional alternatives. The first fictional alternative is
the "worst-case" and has the worst value on every attribute. The
next N fictional alternatives have the worst value on all but one of
the attributes; on the remaining attribute, each alternative has the
best value on one attribute. (Each of these alternatives has a
different best than any of the others.)
3. Rank order the N+1 fictional alternatives. The ranks are determined
by the decision-maker. The rank of the worst-case alternative will
be N+1, and the rank of the best of the fictional alternatives will be
1.
Swing method (Cont.)
4. Rate the N+1 fictional alternatives. The rating of the worstcase alternative will be 0, and the rating of the best of the
fictional alternatives will be 100. The decision-maker must
rate the others and these ratings should be coherent with the
rankings. That is, if one fictional alternatives has a better rank
than a second, the first should have a higher rating as well. An
alternative's rating is the decision-maker's increase in
satisfaction if he gives up the worst-case alternative and
chooses this one instead.
5. Normalize the ratings by dividing each one by the sum of all
the ratings. The normalized rating of the worst-case alternative
will still be 0, and the sum of all the normalized ratings will
equal 1.
6. The weight for each attribute is the normalized rating of the
fictional alternative that has the best value on that attribute.
Source: http://wiki.ece.cmu.edu/ddl/index.php/Swing_weighting