dr. y. İlker topcutopcuil/ya/mdm06mavt.pdfdr. y. İlker topcu () & dr. Özgür kabak...
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Dr. Y. İlker TOPCU
www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info
facebook.com/yitopcu twitter.com/yitopcu
instagram.com/yitopcu
Dr. Özgür KABAKweb.itu.edu.tr/kabak/
http://www.ilkertopcu.net/http://www.ilkertopcu.org/http://www.ilkertopcu.info/http://www.facebook.com/yitopcuhttps://twitter.com/yitopcuinstagram.com/yitopcuhttp://web.itu.edu.tr/kabak/
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MADM MethodsElementary 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 (kabak@itu.edu.tr) 2
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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 iffUi(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 (kabak@itu.edu.tr) 3
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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 (kabak@itu.edu.tr) 4
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MAVT: Value Function Value function is a weighted sum of functions over
each individual attribute:
v(ai) =
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)
n
j
ijjj xvw1
)(
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 5
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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
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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 (kabak@itu.edu.tr) 7
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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
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5
Performance value
Val
ue f
unct
ion
val
ue
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 8
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Exponential Function
Appropriate when performance scores take any value (an infinite number of different values)
For benefit attributes:
vj(xij) =
where is the exponential constant for the value function
otherwise ,
,
)/(exp 1
/)(exp1
*
*
jj
jij
jj
jij
xx
xx
xx
xx
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 9
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Exponential Function For cost attributes:
vj(xij) =
otherwise ,
,
)/(exp 1
/)(exp1
*
*
jj
ijj
jj
ijj
xx
xx
xx
xx
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 10
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Exponential Constant
For benefit attribute
z0.5 = (xm – ) / ( – )
For cost attribute
z0.5 = ( – xm) / ( – )
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))] orTable 4.2. at p. 69 in Kirkwood (1997) is used to calculate R (normalized exponential constant)
= R ( – )
is used to calculate
jx
jx*
jx
jx
jx
*
jx
*
jx
jx
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 11
MDM06Table
http://www.ilkertopcu.info/mailto:kabak@itu.edu.trMDM06Table.jpg
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Exponential Functions
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10
Performance value
Val
ue f
unct
ion
val
ue1
5
5
1
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10
Performance Value
Val
ue f
unct
ion
val
ue
1
5
5
1
Dr. Y. İlker Topcu (www.ilkertopcu.net) & Dr. Özgür Kabak (kabak@itu.edu.tr) 12
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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 (kabak@itu.edu.tr) 13
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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 (kabak@itu.edu.tr) 14
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Values of Global Value Function and Single Dimensional Value Functions
Price Comfort Perf. Design
Norm. w 0,3333 0,2667 0,2 0,2
a 1 0,2705 1 1 1 0,7569
a 2 0,5 1 0,5 1 0,7334
a 3 0,5 0,25 1 1 0,6333
a 4 0,6947 0,25 1 0,2 0,5382
a 5 0,6947 0,25 0,5 1 0,5982
a 6 0,6947 0 1 1 0,6315
a 7 1 0 0,5 0,2 0,4733
v(ai)
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