Dr. Y. İlker TOPCU
www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info
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Dr. Özgür KABAK
web.itu.edu.tr/kabak/
MADM Methods
 Elementary Methods
 Value Based Methods
 Multi Attribute Value Theory
 Simple Additive Weighting
 Weighted Product
 TOPSIS
 Outranking Methods
 AHP/ANP
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
2
MAVT vs. MAUT
 Multi Attribute Value Theory (Evren & Ülengin, 1992; Kirkwood,
1997) –
Weighted Value Function (Belton & Vickers, 1990)–
SMARTS (Simple Multi Attribute Rating Technique by
Swings) (Kirkwood, 1997)
 Multi Attribute Utility Theory (MAUT) is treated
separately from MAVT when “risks” or “uncertainties”
have a significant role in the definition and assessment
of alternatives (Korhonen et al., 1992; Vincke, 1986; Dyer et al., 1992):
 The preferences of DM is represented for each attribute i, by a
(marginal) function Ui, such that a is better than b for i iff
Ui(a)>Ui(b)
 These functions (Ui) are aggregated in a unique function U
(representing the global preferences of the DM) so that the
initial MA problem is replaced by a unicriterion problem.
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
3
MAVT
 This procedure is appropriate when there are
multiple, conflicting objectives and no uncertainty
about the outcome (performance value w.r.t.
attribute) of each alternative
 In order to determine which alternative is most
preferred, tradeoffs among attributes must be
considered:
That is alternatives can be ranked if some procedure
is used to combine all attributes into a single index of
overall desirability (global preference) of an
alternative:
A value function combines the multiple evaluation
measures (attributes) into a single measure of the
overall value of each alternative
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
4
MAVT: Value Function
 Value function is a weighted sum of functions over
each individual attribute:
n
v(ai) =
 w v (x
j 1
j
j
ij
)
 Thus, determining a value function requires that:
 Single dimensional (single attribute) value functions
(vj) be specified for each attribute
 Weights (wj) be specified for each single dimensional
value function
 By using the determined value function preferences
can be modeled:
a P b  v(a) > v(b);
a I b  v(a) = v(b)
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
5
Single Dimensional Value Function
 One of the procedures used for determining a single
dimensional value function that is made up of
segments of straight lines that are joined together
into a piecewise linear function,
 while the other procedure utilized a specific
mathematical form called the exponential for the
single dimensional value function
v(the best performance value) = 1
v(the worst performance value) = 0
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
6
Piecewise Linear Function
 Consider the increments in value that result from
each successive increase (decrease) in the
performance score of a benefit (cost) attribute, and
place these increments in order of successively
increasing value increments
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
7
Value function value
EXAMPLE: 1-5 scale for a benefit attribute
Suppose that value increment between 1 and 2 is twice as
great as that between 2 and 3. Suppose that value
increment between 2 and 3 is as great as that between 3
and 4 and as great as that between 4 and 5. In this case
piecewise linear single dimensional value functions would
be:
v(1)=0, v(2)=0+2x, v(3)=2x+x, v(4)=3x+x, and v(5)=4x+x=1
v(1)=0, v(2)=0.4, v(3)=0.6, v(4)=0.8, and v(5)=1
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5
Performance value
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
8
Exponential Function
 Appropriate when performance scores take any
value (an infinite number of different values)
 For benefit attributes:
vj(xij) =
1  exp  ( x  x  ) /  
ij



j
,  

*

 1  exp  ( x  x )/ 
j
j



 x  x
j
 ij
, otherwise
 *

 x j  x j
where  is the exponential constant for the value function
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
9
Exponential Function
 For cost attributes:
vj(xij) =
1  exp  ( x   x ) /  
ij



j
,  


*
 1  exp  ( x  x )/ 
j
j



 x  x
ij
 j
, otherwise
 
*
 x j  x j
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
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Exponential Constant
 For benefit attribute
z0.5 = (xm – x ) / ( x* – x j )
j
j
 For cost attribute
z0.5 = ( x  – xm) / ( x – x* )
j
j
j
are used (where xm is the midvalue determined by DM such that
v(xm)=0.5) to calculate z0.5 (the normalized value of xm)
 The equation [0.5 = (1 – exp(–z0.5 / R)) / (1 – exp(–1 / R))] or
Table 4.2. at p. 69 in Kirkwood (1997) is used to calculate R
(normalized exponential constant)
MDM06Table
  = R ( x* – x  )
j
j
is used to calculate 
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
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Value function value
Exponential Functions
1
1
5
0.8
0.6
0.4
5
1
0.2
0
0
1
2
3
4
5
6
7
8
9
10
Value function value
Performance value
1
1
5
0.8
0.6
0.4
5
1
0.2
0
0
1
2
3
4
5
6
7
8
9
10
Performance Value
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
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Example for MAVT
 Price: Exponential single dimensional value function
 Other: Piecewise linear single dim. value function
 Let the best performance value for price is 100 m.u., the worst
performance value for price is 350 m.u., and the midvalue is 250
m.u.:
 z0.5=0.4  R = 1.216   = 304
 vp(300)=0.2705, vp(250)=0.5, vp(200)=0.6947, vp(100)=1
 Suppose that value increment for comfort between “average” and
“excellent” is triple as great as that between “weak” and “average”:
 vc(weak)=0, vc(average)=0.25, vc(excellent)=1
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
13
Example for MAVT
 Suppose that value increment for performance between
“weak” and “average” is as great as that between “average”
and “excellent”:
 va(weak)=0, va(average)=0.5, va(excellent)=1
 Suppose that value increment for design between
“ordinary” and “superior” is four times as great as that
between “inferior” and “ordinary”:
 vc(inferior)=0, vc(ordinary)=0.2, vc(superior)=1
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
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Values of Global Value Function and
Single Dimensional Value Functions
a2
Price
0,3333
0,2705
0,5
Comfort
0,2667
1
1
Perf.
0,2
1
0,5
Design
0,2
1
1
a3
0,5
0,25
1
1
0,6333
a4
0,6947
0,25
1
0,2
0,5382
a5
0,6947
0,25
0,5
1
0,5982
a6
0,6947
0
1
1
0,6315
a7
1
0
0,5
0,2
0,4733
Norm. w
a1
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak ([email protected])
v(ai)
0,7569
0,7334
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