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Analyzing the Problem (MAVT)

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 .

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Analyzing the Problem (MAVT)

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  1. Analyzingthe 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 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.

  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: Avalue functioncombines 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(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)

  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: vj(xij) = where is the exponential constant for the value function

  9. Exponential Function • For cost attributes: vj(xij) =

  10. 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 

  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.:  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

  13. 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

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

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