© A. Schwing, 2010Institute of Visual Computing Informatik I for D-MAVT Exercise Session 10.
Analyzing the Problem (MAVT) Y. İlker TOPCU, Ph.D. twitter.com/yitopcu.
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Transcript of Analyzing the Problem (MAVT) Y. İlker TOPCU, Ph.D. twitter.com/yitopcu.
Analyzing the Problem(MAVT)
Y. İlker TOPCU, Ph.D.
www.ilkertopcu.net www.ilkertopcu.org www.ilkertopcu.info
www.facebook.com/yitopcu
twitter.com/yitopcu
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)