Analyzing the Problem(MAVT)

Y. İlker TOPCU, Ph.D.

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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 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.

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

jijjj xvw

1

)(

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

Piecewise Linear Function

EXAMPLE: 1-5 scale for a benefit attributeSuppose 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=1v(1)=0, v(2)=0.4, v(3)=0.6, v(4)=0.8, and v(5)=1

00.20.40.60.8

1

1 2 3 4 5

Performance value

Val

ue f

unct

ion

valu

e

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))] or Table 4.2 at p. 69 in Kirkwood (1997) is used to calculate R (normalized exponential constant)

• = R ( – ) is used to calculate

j

x j

x*

jx

j

x j

x *

jx

*

jx

jx

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

valu

e

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

valu

e

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 acceleration 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. DesignNorm. w 0,3333 0,2667 0,2 0,2a 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)