dr. y. İlker topcutopcuil/ya/mdm06mavt.pdfdr. y. İlker topcu () & dr. Özgür kabak...

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/

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 ([email protected]) 2

http://www.ilkertopcu.info/mailto:[email protected]

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.



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



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

)(



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



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



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



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



Exponential Function For cost attributes:

vj(xij) =

otherwise ,

,

)/(exp 1

/)(exp1

*

*

jj

ijj

jj

ijj

xx

xx

xx

xx



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


MDM06Table


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



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



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



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)


dr. y. İlker topcutopcuil/ya/mdm06mavt.pdfdr. y. İlker topcu () & dr. Özgür kabak...

Documents