1. 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

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 U i, such that a is better than b for i iff U i (a)>U i (b) These functions (U i ) 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.

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

4. MAVT: Value Function Value function is a weighted sum of functions over each individual attribute: v(a i ) = Thus, determining a value function requires that: Single dimensional (single attribute) value functions (v j ) be specified for each attribute Weights (w j ) 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)

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

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

7. Piecewise Linear Function 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

8. Exponential Function Appropriate when performance scores take any value (an infinite number of different values) For benefit attributes: v j (x ij ) = where  is the exponential constant for the value function

9. Exponential Function For cost attributes: v j (x ij ) =

10. Exponential Constant For benefit attribute z 0.5 = (x m – ) / ( – ) For cost attribute z 0.5 = ( – x m ) / ( – ) are used (where x m is the midvalue determined by DM such that v(x m )=0.5) to calculate z 0.5 (the normalized value of x m ) The equation [0.5 = (1 – exp(–z 0.5 / 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 

11. Exponential Functions

12. 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.:  z 0.5 =0.4  R = 1.216   = 304  v p (300)=0.2705, v p (250)=0.5, v p (200)=0.6947, v p (100)=1 Suppose that value increment for comfort between “average” and “excellent” is triple as great as that between “weak” and “average”:  v c (weak)=0, v c (average)=0.25, v c (excellent)=1

13. Example for MAVT Suppose that value increment for acceleration between “weak” and “average” is as great as that between “average” and “excellent”:  v a (weak)=0, v a (average)=0.5, v a (excellent)=1 Suppose that value increment for design between “ordinary” and “superior” is four times as great as that between “inferior” and “ordinary”:  v c (inferior)=0, v c (ordinary)=0.2, v c (superior)=1

14. Values of Global Value Function and Single Dimensional Value Functions